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Complex Systems - Jean-Philippe Bouchaud - Lecture 2: Multiplicative Growth (I). Concentration — Transcript

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0:00this conference will now be recorded
0:07Okay so
0:09after this initial delay I can
0:12continue
0:14telling you about
0:16Central limit theorems and so I haven't
0:19had time to write my the outline uh
0:22because of these technical problems
0:24but I guess is going to be okay
0:27so I I was talking last time about the
0:31central limit theorem and its
0:33generalizations
0:36foreign
0:43is that when you have parallel
0:45distributed variables
0:48with a sale that I
0:51usually write like this then when mu is
0:56greater than 2
0:57is the usual
1:00CLT
1:03with all the provisos that we mentioned
1:06that you have to use it in the correct
1:08region and this region might depend on
1:10the problem and so on
1:12but when U is less than two then
1:16something different happens
1:18and in particular what happens is that
1:21the rescaling is not the same as usual
1:23in the sense that SN if you start from
1:28again even distribution not to care
1:30about the mean so
1:33I'm going to think of a problem with a
1:35mean
1:36is zero just for Simplicity then what we
1:39saw is that the order of magnitude of SN
1:42SN being
1:45the sum of the x i
1:49is not square root of n that is n to the
1:52one over mu
1:55and I ended up last time saying
1:59we've encountered this n to the power 1
2:01over mu before
2:02it is actually exactly the same order of
2:05magnitude
2:06as MN
2:09which is
2:12the maximum over uh all these random
2:16variables of x i
2:19and so this means that
2:22in a sense the sum is dominated by its
2:25largest terms okay
2:27and this is very different from what
2:29happens in the usual elt case
2:32where SN is over the square root of n
2:35and when mu is greater than 2 square
2:38root of n is much larger than M than n
2:40to the power 1 over U so PSI I I
2:45name this situation Democratic in the
2:48sense that all the variables contribute
2:50more or less equally and here we're in a
2:53case where
2:54this democracy is uh stating and a few
2:58actors play a major role
3:01so that's what I want to tell you about
3:04today is about something that people
3:07call
3:08concentration
3:13it's also called localization
3:20and I'll mention
3:23that in some physical context
3:25it is also associated with something
3:28that people call freezing
3:31okay so what I want to talk about now is
3:35how to characterize this thing that I
3:38said
3:39in a hand waving way that is that the
3:42sum is concentrated in a few of its
3:46members of its constituents
3:49and so I'm going to introduce uh
3:53the concentration indicator
4:02foreign
4:04which happens to have been introduced in
4:07various names in different contexts
4:12so for example in physics it's called
4:14the inverse participation ratio
4:25and in economics it's called the central
4:27index
4:34and it still has other names in the
4:37ecology and and other disciplines but so
4:41so this this idea that I'm going to talk
4:43about you you actually is going to be
4:45related to something that you know very
4:48well
4:49um
4:49but it is something that has been
4:52introduced in many different contexts
4:55so in order to keep the the
4:58discussion as simple as possible I'm
5:01going to
5:03focus on variables that are
5:07non-negative
5:11and actually a lot of my examples later
5:14in the lecture
5:15are going to be is going to be a lot of
5:19my examples are going to be devoted to
5:21these uh
5:23these positive random variables like
5:25wealth or a number of species or keep a
5:29number of population in a city and so on
5:31okay so so X is a positive variable and
5:36I'm going to associate to x i
5:40the weights wi which is the weight of x
5:45i in the sum and so wi is by definition
5:49x i over s n
5:52okay
5:54so because all the X i's are positive
5:56all these weights are also positive or
5:58zero and clearly by definition sum over
6:03I
6:04of w i is 1.
6:06so these are really weights
6:10so let me see uh until when I write here
6:13you see okay I should stop here
6:19okay so now from this way I'm going to
6:23introduce a family of these
6:26concentration indicators
6:28and one special member of this family
6:30will be the half pencil index or the
6:32in-house participation ratio but you you
6:34in principle all the members of the
6:37family can be important too and one of
6:40them uh as physicists you already know
6:43about
6:45so I'm going to introduce these are
6:46simple index
6:48uh Cody H index Q so there's a
6:53an index queue which
6:56parameterize the summary and it's going
6:59to be equal to the sum over I of w i
7:03to the Q
7:05and Q is going to be
7:07positive or zero
7:10or actually they're interesting if
7:12they're Plus or equal to one
7:16Okay so
7:18the standard half window index
7:21is
7:22Q equal to
7:26for Q equal 1
7:28then obviously because of the
7:31normalization H is always equal to one
7:34H1 is always equal to one
7:37and uh if Q
7:40is equal to one plus Epsilon
7:43with Epsilon going to zero
7:46then
7:47by a very simple manipulation uh of of
7:53this sum you see w i to the 1 plus
7:55Epsilon I'm going to write as exponent
7:59of w i well let me write it down so if I
8:02write w i it's the one that's Epsilon
8:04equals w i
8:07Flex Financial of excellent log w i
8:13then what I find is that h
8:16UE
8:18is the first order in epsilon equal to
8:22one
8:24plus Epsilon
8:26sum over I of wi log
8:30of WI
8:33and most of you I guess know that this
8:36object here if you have
8:39weights that come to one I.E
8:41probabilities then this object is minus
8:45the entropy of the W's
8:50and again this as I'm going to show for
8:54a half into index you already see that
8:57this is going to be a measure of
9:00concentration we know that entropy
9:03measures how the measure how the measure
9:07how the probability over phase space you
9:10know thermodynamical system spreads out
9:13overall possibilities
9:16if you start from a well-defined initial
9:19condition
9:20the probability
9:22is localized around this this initial
9:25condition and the entropy is small and
9:28then we know
9:30by the way I need to remove this because
9:35Like Oxygen
9:45Okay so
9:48so we know that entropy is growing with
9:51time and this is associated with the
9:53fact that as time goes on
9:55to the probability distribution that I'm
9:58going to you know draw in a kind of
10:00abstract Faith phase starts off
10:02localizes around an initial condition
10:04and then as time goes on it spreads out
10:09and it becomes more and more uniform
10:11over face space at least in the
10:14microeconomic canonical ensemble
10:17and so indeed HQ when Q goes to to 1 is
10:22associated to something that measures
10:24how concentrated how or how uniform
10:27uh the WIS are
10:31and it's the same for
10:33all values of Q
10:36greater than one
10:38and in particular
10:40let me Focus
10:42on H2
10:44H2 which is the sum
10:47over I of w i squared
10:51and so imagine for example that the wi's
10:54Are all uh equal to one over n
10:59so this is the
11:01complete uniformity and no concentration
11:05at all then you find that H2
11:10is the sum Over N terms that are equal
11:14and they're all equal to 1 over N
11:15squared so it's 1 over n
11:19and this goes to 0 as n goes to Infinity
11:23so this indicator this is a central
11:26index goes to zero when
11:29the weights are equally distributed
11:32but imagine on the country that
11:35one of the wi wi zero say is a fixed
11:40number a
11:42and then all the other ones wi different
11:45from i0 is equal to 1 minus a times 1
11:50over n minus 1.
11:53okay so in this case we would have
11:56uh
11:57weights that are concentrated around one
12:00particular value of I
12:03and uh
12:05uniform elsewhere
12:07then you see that well I've actually
12:10defined it in such a way that the sum of
12:12the wi is equal to one
12:15but if I compute H2 in this case
12:22and take the limit
12:24n to Infinity
12:26I find that it's equal to a squared
12:31which does not go to zero uh when
12:36uh
12:41when um n goes to Infinity
12:45so that's the the nice property about H2
12:49or actually
12:52other values of Q is that it's a
12:55it's an indicator that for very large
12:57systems it is either very small if the
13:01weights are well spread out or remains
13:04of other one if there's concentration
13:07and so I'll speak about concentration in
13:10this case the concentration
13:20is when the limits when n goes to
13:23Infinity
13:24of H2
13:26is uh
13:29strictly positive
13:33okay
13:36so let's go back to the central limit
13:38theorem
13:40and ask about the the values of H2 in
13:44the different cases
13:46and so when mu
13:49is greater than 2
13:51not surprisingly because of what I said
13:54earlier that SN is square root of n
13:57and it's much bigger than any of the
14:00random variables because the largest one
14:02is much smaller is n to the one over mu
14:05which is much less
14:07in square root of n if mu is greater
14:10than 2 then what you find is that h
14:15and by the way from now on I'm going to
14:17drop index 2 I'm going to call H the
14:21half Intel index by default this is Q
14:24equal 2.
14:25it's of order one of red
14:30and it goes to zero when n goes to
14:32Infinity
14:35when mu is less than one
14:39on the other hand H is over the one
14:44when n goes to Infinity
14:49and so we are in a situation that I call
14:53concentrated
14:54and again intuitively it's clear that
14:59the largest term has a finite weight
15:01because it's of the same order of
15:03magnitude as the whole sum and so we
15:06will be in a situation a little bit like
15:08this actually it's a little more
15:10complicated than this
15:11but you understand already from the
15:14scaling that we're in a concentrated
15:16situation
15:19by the way there's something interesting
15:22to mention here
15:24is what do I mean by of the order of in
15:27this case
15:30well you see that H2 is itself
15:34the sum of a very large number of random
15:39variables okay
15:40the WIS are random because dxis are
15:43random
15:45and so here I'm summing many random
15:47variables that are all actually less
15:50than one
15:51and so you might naively think that
15:54there's some kind of central limit
15:55theorem
15:56or at least a low of large number that
15:59applies to this sum
16:01but when mu is less than one this is not
16:03the case
16:05what happens is that h
16:08continues to fluctuate does not
16:15converge
16:18to a number
16:25so what I mean by this is that although
16:27you sum a very large number of terms
16:30this concentration indicator is always
16:34over the one but it's going to vary from
16:37one sample to the next
16:39so what I call a sample is a particular
16:42choice of X1 X2 xn
16:46and for each of these samples you'll get
16:49a different value of H
16:51so it's it's a strange situation which
16:54often is is called non-self averaging so
16:58one sums a very large number of items
17:02but still never comes largest to
17:04anything
17:05and so if I now plot the distribution of
17:08H
17:09since it doesn't converge it must be it
17:12must have a
17:13non-trivial distribution and it indeed
17:16has a very funny shape
17:19so H is between 0 and 1.
17:22I failed to mention this but it's very
17:25easy to show that
17:26this is
17:29between 0 and 1.
17:32and the distribution as as I said a very
17:35exotic shape it diverges close to
17:40to Mu to H equal one then it has a
17:43little bump at one half then another
17:46Singularity at one third and so on
17:49so it looks like this
17:51the distribution depends on mu and in
17:54particular the average value of H
17:56so the average value of this
17:58distribution is equal to 1 minus mu
18:03okay
18:04so when mu goes to one
18:08we leave this extreme concentration
18:10regime and
18:12the value of the average value of H goes
18:15to zero and the whole distribution
18:17actually moves kind of zero
18:20whereas when mu goes to zero
18:24that is for really extremely broad
18:26distributions
18:27the average value of H goes to 1 which
18:30means that a single term always
18:33dominates the whole sum
18:36Okay so
18:38we have a concentrated situation here or
18:41localized localized on a few random
18:44variables
18:45we have a completely delocalized
18:47situation
18:49in the usual Central limit ethereum case
18:52so what happens in the middle and so
18:54you've seen the back I've carefully
18:58left space for the intermediate case
19:00which happens to be uh
19:03much more subtle
19:06in the sense that
19:08the typical value of H
19:11I'm going to write a typical
19:14which means for example the median value
19:16the value such that there are half of
19:19the samples that are that have an h less
19:21than that and half that have an H
19:23greater than that
19:25so the typical value is of order n
19:28to the 2 1 minus mu
19:32over mu
19:35whereas the average value of H
19:40is dominated by Rare samples
19:43and is of older
19:47n to the 1 minus B
19:51which is much greater
19:53than hypical
19:55foreign
20:05and if you want to try to do it I think
20:08it's a it's a nice exercise to try to
20:11think of how you would show this for
20:13example how would you show that the
20:15typical value of H is of disorder of
20:18magnitude this is not terribly
20:20complicated with a tools I've given you
20:23already I mean the hand waving tool but
20:25I think you can do that it's a little
20:27more complicated to show this but if
20:30you're interested we could discuss that
20:33how to
20:34plant the problem in such a way that you
20:36can actually compute this exactly
20:38including the the prefactor
20:41so this is not really concentrated
20:45because H goes to zero but it's not
20:48really delocalized either because uh H
20:51is not
20:53of older one of red one over N means
20:55that it's really well spread out over
20:57all sides so there's a kind of
21:00pre-concentration effect here
21:02but it's not fully developed okay
21:09so we'll see an example
21:12of
21:13this phase transition as it were between
21:16a delocalized phase and a localized
21:19phase in the example I'm going to
21:21discuss
21:23in a few minutes
21:25but you see that there's an interesting
21:27phenomenology coming out here and I
21:30would have loved to show you an example
21:32of that a visual example of that but
21:35unfortunately uh
21:37well because we had to change computer
21:40my file is not on the computer anymore
21:42maybe you have it make
21:44yeah of last time can you share the
21:46screen
21:51so maybe we can
21:55do it thanks to
22:28yeah okay if it's too complicated
22:33I'll I'll show you next time I'll start
22:35by that next time
22:42okay don't worry we'll do it next time
22:47so do you have any question at this
22:49stage
22:57no don't worry we'll we'll do it next
22:59time
23:02Okay so this is a concentration
23:05indicator there are many different ways
23:07to characterize inequalities
23:10um so you see that
23:12inequality is what age measures
23:16so mu greater than 2 means that there's
23:18very little inequality U lesson one that
23:21there's huge inequality
23:23and before moving on to the next object
23:25I want to mention another very popular
23:29measure of inequalities especially in
23:31economics
23:33which is called the genie coefficient
23:36all right
23:41so I'm not going to go into much detail
23:43but just want to tell give you the
23:45definition of the genie coefficient
23:47which is another natural way of
23:50measuring how an equal uh the excise are
23:54and it's it's famously used to
23:57characterize inequalities of wealth or
24:00in of income around the world so you'll
24:03find on Wikipedia for example tables of
24:07Genie coefficients measuring
24:09inequalities in different countries of
24:12the world and the genie in the genie
24:14coefficient G
24:16is defined as one over two n
24:21um over all path I and J of x i minus x
24:26j
24:27absolute value divided by
24:30sum over I of x i
24:34so what you're doing here is roughly
24:37speaking you're comparing the average
24:39distance between two randomly chosen
24:42individuals
24:44I mean the wealth of two randomly chosen
24:47individuals or their income to the
24:49average okay so this is an a dimensional
24:52object G
24:55as it should be of course H also is an a
24:57dimensional object
24:59doesn't depend on the currency in which
25:02you're measuring well for example
25:03obviously and it's a number again that's
25:06between zero and one
25:09oops I shouldn't go too far here
25:16and so um
25:18if it's zero it means that everybody is
25:20equal and if it's one it means that one
25:23guy is actually uh dominated dominating
25:26the whole sum so it's it's really very
25:29close in spirit to the final coefficient
25:32and they're having no index has
25:34different properties though
25:36uh and I I won't go more in details but
25:41you can really see uh the genie
25:44coefficient as
25:45another definition that's a member of a
25:49bigger family and in this family there's
25:51also the half handle index so I won't go
25:53into that but essentially these are
25:56different measures of inequalities that
25:58are routinely used
26:01so just to mention that if you look at
26:04wealth inequalities uh around the world
26:07uh the genie coefficient is of order
26:100.7
26:13so for many countries in the world
26:18which is pretty high
26:20and as we we know it has increased over
26:23the last uh decades you see the maximum
26:27value is one so 0.7 is is already large
26:30it fluctuates of course from one country
26:32to the next or it varies from one
26:35country to the next and in the US for
26:37example it's even larger than that
26:41okay
26:45so
26:48before moving to one specific example
26:53of
26:55parallel random variables and uh
26:59anomalous Central limited theorems
27:03I want to mention something very general
27:05at this stage
27:07which is that
27:10I've given you one example where the
27:12central limit theorem is violated which
27:15is the case where
27:17variables lose their second moment okay
27:21so that's one example where obviously
27:23the central limit theorem has to be
27:25amended
27:27but what's surprising is that actually
27:30the central limit theorem is extremely
27:32robust
27:33so I told you at the beginning
27:35last week
27:37when I started presenting the central
27:39living theorem that usually it is proven
27:42under the assumption that
27:44you have rid random variables
27:48so independent identity distributed but
27:52you can add quite a an amount of
27:55correlations for example between these
27:58variables or you can make them
28:01non-identically distributed and in many
28:04cases you're not able to break the
28:07central limit theorem
28:09you're only able to break the central
28:10limit theorem in some kind of extreme
28:13cases one of these extreme cases is the
28:16one I just discussed the one where you
28:18lose the second moment it's pretty
28:19extreme but it happens
28:22and other cases are for example when the
28:25exercise that you sum have very long
28:28range correlation
28:30okay so if the correlation between the X
28:33size so imagine that the excise are
28:35drawn
28:36one after the other
28:38if there's a a small autocorrelation in
28:41time of these random variables then as I
28:45said you're not going to break the
28:46central limit theorem still going to be
28:48the option
28:49but if there's long range correlations
28:51if these coalitions Decay very slowly
28:53actually as a parallel of time
28:56then
28:57in some cases you also break the central
29:00limit theorem
29:02so
29:03if you want there's there are
29:05two broad mechanisms for breaking the
29:09central limit theorem one is
29:12anonymously anomalously large fat tail
29:17and the other one is a very long range
29:21correlations
29:23and uh and in these cases you have to do
29:27something different
29:29okay
29:31so let me now move to
29:34um
29:35the main subject of today
29:39which is
29:41multiplicative growth
29:45yes
29:49so Krishan is asking can someone explain
29:51P of H as a function of age
29:55distribution of the index
30:00sorry uh
30:09can I can one derive this this form
30:13okay so actually it is there's no
30:17analytical form for this P of H I'm just
30:20drawing it so you can of course compute
30:22it numerically what you can do is to
30:26characterize the singularity so there's
30:28a Divergent Divergence here close to Mu
30:30one there's a singularity equal to uh to
30:34sorry to H equal one there's a
30:36singularity equals to H equals one half
30:38there's another thing like the close to
30:41all these
30:43simple fractions one third one fourth
30:46and so on and one can characterize
30:49the the the strength of these
30:51um
30:52of these singularities but the full
30:55function you see it's such a strange
30:57object
30:58that there's no closed formula
31:02by the way these fractions here one
31:05one-half one-third one-fourth and so on
31:09um they have a special
31:12um interpretation
31:13so in the sense that if you have exactly
31:18small n objects that have a weight
31:21wi equal 1 over n
31:24and all the other ones
31:26have a weight
31:27zero
31:29then in this case you find that h
31:32is equal to one of rent
31:36okay
31:37so if you break
31:39the sum into equal pieces
31:43there's something singular happening
31:45that's that's the interpretation of
31:47these singularities they you happen to
31:49have a kind of special probability for
31:52having an exact breakdown in equal parts
31:55okay but of course there's no more than
31:59than that that you can say about this
32:01function
32:07[Music]
32:28thank you
32:30okay so the topic of today really I mean
32:33there's a kind of second chapter
32:35of the lecture Is Random
32:41multiplicative growth
32:51foreign
32:54by the way my screen is exhausted but I
32:57guess that you see the okay good because
33:00in my screen I see from right to left
33:04um Okay so
33:06so what's the motivation here so I give
33:08you a long introduction about power laws
33:11power distributions
33:13this exponent mu that I keep calling mu
33:17and
33:19they seem to pop up in many different
33:22situations
33:23and one would like to understand the way
33:25where do they come from
33:28what do they tell us about the
33:29underlying system
33:31and if you have parallels
33:33well very often they're Associated to
33:37the free transition they're Associated
33:39to something really very non-trivial
33:42happening in the system where the system
33:44kind of hesitates between two macro
33:47state
33:48if you think of magnets for example then
33:51you you know that there's a high
33:53temperature phase where there's no
33:55magnetization and the low temperature
33:57phase where there's some magnetization
33:59and right at the critical point where
34:02the system kind of doesn't know where it
34:04where to go then you have all these very
34:08interesting phenomenon of
34:10scaling variants
34:12and Associated parallels you remember
34:14last time
34:16I gave you the link between parlors and
34:19scale endurance
34:21but it's not always the case in some
34:24cases parallels can arise through
34:27simpler mechanisms
34:29and one of these mechanisms that I'm
34:31going to explain today before moving to
34:34these the phase transition examples
34:36later in the lecture is random
34:40multiplicative growth and you'll see
34:42that this is a
34:44basic setting framework that allows
34:47already to cover many interesting
34:49phenomena
34:50like population growth wealth growth
34:54uh spot market growth and so on and so
34:59I'm going to tell you about this model
35:02in more detail now by the way
35:05um I guess that you've had you received
35:08the first chapter
35:09uh the of the lecture notes as I
35:13mentioned this is the in preliminary
35:16state I'm rushing to have as many
35:19chapters written but
35:21don't you know don't think that it's
35:23going to be all nice and ready before
35:25the end of this particular
35:27uh school year
35:30um there's a lot of
35:32typos and problems in in these notes I'm
35:36sure so please uh give feedback if you
35:39can but I just wanted to point out
35:41something that I didn't say yet is that
35:43I'm I'm trying to break these chapters
35:46into parts that I consider to be less
35:49important and part that are more
35:51important and the less important part
35:54more technical Parts there's a gray bar
35:57on the left so you don't need to read
36:00everything that's that I've written in
36:02these notes
36:04um there's a mixture of more advanced
36:06stuff and more Elementary stuff and so
36:09you can choose between the two using the
36:12code bar
36:14we need to code the
36:17left bar Okay so
36:21what am I what am I going to try to
36:24describe here let me State the problem
36:27in terms of City growth
36:30and I told you
36:32cities have a very broad distribution it
36:35has in in terms of their size
36:37it's a zip flow it's one of the oldest
36:40example
36:41of a parallel distribution
36:44which means that there are big cities
36:46and much smaller cities
36:49and so I'm going to call I the index of
36:52a city they are capital N cities in the
36:55in the country
36:58and I'm going to call Zi
37:01the total population
37:08in CTI
37:11okay
37:14and I'm going to try to model
37:17uh the the evolution of that eye in time
37:24and the way I'm going to model it is to
37:26postulate that
37:30the evolution of that I as a function of
37:33time is it IDT
37:35is a term proportional to z-i that I'm
37:38going to call m i z i
37:43so this is proportional growth
37:46the more individuals you have in a city
37:49the more likely they have to they they
37:51are to have kids and therefore the
37:55fastest the faster the growth of the
37:57population
37:58and so mi is the uh average
38:03reproductive reproduction rate
38:12okay
38:16but I'm going to also try to model the
38:19fact that
38:20in some circumstances
38:22because the population is more wealthy
38:26or less wealthy or because there are
38:28special situations meaning that newborns
38:32have a lower probability of surviving or
38:36higher probability of surviving so this
38:37can be due to
38:39uh you know climate changes or or
38:42viruses or whatever so I'm going to add
38:45some Randomness here so what I'm going
38:48to say is that overall in the city
38:51there's a random term that I'm going to
38:54call Eta I of t
38:56times z i
38:59which
39:00which means that
39:02even if on average over time
39:05the reproduction rate is is MI there are
39:08fluctuations in this reproduction rate
39:10which as I said can depend on many
39:13different factors and I'm throwing all
39:15these factors into a random term
39:19that I'm calling h i
39:25have actually another term here that I
39:27should write
39:28that I'm going to discuss in Great
39:30Lengths in the next lectures and so I'm
39:34putting it here just uh to uh remember
39:39it and not completely sweep it under the
39:42rank but there's actually a term that
39:45comes from the fact that even if the
39:47average reproduction rate is fixed then
39:50from one year to the next maybe people
39:53randomly have more children or less
39:55children and so there's a term
39:57proportional to square root of z i times
40:01another random variable PSI
40:04but I'm just you know flashing it and
40:07I'll explain much better where this sum
40:09comes from later on I'm not going I'm
40:12going to completely neglect this term
40:14for the present discussion
40:16and one of the uh reason to do that is
40:20that
40:21when set is large
40:24uh square root of that is much less than
40:26said and so I can safely neglect that
40:28term
40:30why why am I interested in large Zeds
40:34well uh because the
40:38one of the reasons is that I'm going to
40:40be interested in the tail the Tails of
40:42the distributions and then my my
40:44definition the tails are when the when
40:47set is large
40:48okay
40:50so that's the model I'm going to
40:52consider of course I need to specify
40:55what this random term here is and this
40:58is going to take a little a little bit
41:01of our time
41:02because there's there's subtlety in the
41:04description of this ATI but before doing
41:07this I want to insist on the fact that
41:10I've spoken about cities and population
41:13but you can think of this model in many
41:15different other contexts so for example
41:18you can think of I being an individual
41:29individual
41:31and said I as it as well
41:36okay
41:40in this case Mi would be the average
41:44return on the wealth of individual I and
41:47so I don't know some smart people may
41:50have a larger m in vice versa and then
41:55you know also obviously depending on
41:58what the stock market is doing for
42:00example or the state of the economy even
42:02if you have an average rate of return on
42:05your Capital then you expect to also
42:07have some fluctuation some years are
42:10good some years are bad and that's what
42:13the random term tries to capture
42:17you can think of I a species
42:21different species in an ecological
42:24system and Zed as the population of that
42:29species the number of
42:31individuals of that particular type
42:34and so again instead of countries I have
42:37species and I have the same logic for
42:40writing down such an equation
42:42and so on and so forth
42:45so this model can represent many
42:49different
42:50uh situation
42:53okay but mostly I'm going to you know
42:56tell you the story either in terms of
42:58cities or in terms of of wealth but you
43:00can adapt
43:02uh the language to any problem that you
43:05want to speak about
43:07okay so now I need to spend some time on
43:10this at I of T because uh immediately
43:13something is going to pop up
43:15which is how do I solve this
43:19uh differential equation which contains
43:22a random term
43:25so I'm going to assume that ETA
43:29are
43:30random
43:32and the time dependence of this
43:34randomness
43:36is
43:40such that so the brackets mean average
43:42value over the realization of the
43:45process
43:46at different times
43:47this correlation function so this is the
43:50correlation function of the noise ETA
43:53sorry I should have said something right
43:55away which is that I'm assuming that
43:57this as a zero mean
44:06the average value of ETA is zero
44:10clearly I've put everything in the mean
44:13here Mi so the residual is something
44:18that has zero
44:19so this is the correlation function of
44:21the noise
44:23and I'm going to assume that it has this
44:26this shape Sigma squared over 2 Tau C
44:29exponential of minus t minus P Prime
44:33of its houses
44:37so if I plot this correlation function
44:39as a function of the difference T minus
44:42t Prime
44:47and it starts at some value and it
44:49decays exponentially over a time scale
44:52that is Tau key
44:55so Tau C is the correlation time of the
44:57noise
45:05and if you look at time 0
45:08then time 0 is T minus D Prime equals 0
45:12in lag zero that should be a lag
45:16the lag between TNT Prime
45:19at lag zero this is zero so it's Sigma
45:22squared
45:23over two thousand
45:27and what is this value at black zero
45:30well it's it's simply the variance of
45:33the Noise Okay so this is the variance
45:41so this form here means that you have a
45:44random process that's Auto correlated in
45:47time
45:48but it quickly loses its memory and
45:52after Tau C it essentially doesn't
45:54remember what it did in the past
45:58so coming back to my example of
46:01populations you can think for example of
46:04uh
46:06Health situations or crops or whatever
46:11giving rise to these fluctuations in
46:13reproduction rate and so
46:16if the health situation deteriorates for
46:19some years and then improves again for
46:21some years how she will be of older
46:24years
46:25uh and it it's not necessarily true that
46:28it's going to be exactly of that form
46:30but that's the way I want to think about
46:32this this noise okay
46:35so now I want to
46:37give you a little story about
46:40this random differential equations
46:46that depends on what should
46:49choose Tau C to be
47:07foreign
47:10so often in physics
47:12we think of you know the nature to be
47:16not to have any
47:20um infinitely short correlation time we
47:23always think that there is some cutoff
47:24in any phenomenon that you might
47:27reasonably consider so usually in
47:31physics papers palsy
47:34is uh often considered to be finite it
47:38may be small
47:40but it's not zero it's not infinitely
47:43small
47:44and so if Tau T is greater than zero
47:48then ETA is a random function but it's a
47:51regular random function
47:54so if I plot as a function of T A
47:57Certain realization of ETA is going to
48:00look messy it's going to look random
48:02but it's smooth
48:05and I'm trying to on my plot here make
48:08obvious that this is tausi okay
48:15and if I have a you know random function
48:17that is smooth I can treat
48:21uh the differential equation that I've
48:24written up there as a kind of it's not a
48:26kind of it's an auditory ordinary
48:28differential equation
48:31and if it's an ordinary differential
48:33equation I can do what I usually do with
48:35differential equations I can for example
48:37change variables so imagine that I
48:41introduce U equals log Z
48:47okay then g u d t
48:52is D log Z DT which is one over Z
48:55DP
48:59and
49:00one of that is that DT is m
49:06plus Theta of t
49:10so I'm dropping the I here
49:15foreign
49:30so now I have a very simple differential
49:34equation u d t equals M plus h of T and
49:38therefore U of T
49:42is U of 0
49:44Plus Mt
49:46plus the integral from 0 to T of ETA of
49:50T Prime
49:51DT Prime
49:57so I'm going to come back to that later
50:04interprets what I'm finding here but
50:08before doing that I want to give you
50:10another path that you could follow
50:23which leads to a different
50:29IAL equation for U
50:37that can be justified in some cases but
50:41what I want to say here is beware
50:44because there's a there's this famous
50:47problem between the interpretation of
50:50such
50:51uh random differential equation is that
50:55you could you could think of
50:58healthy to be Zero from the start
51:04actually
51:06more precisely because there's an
51:09infinite time scale Infinity small time
51:12scale DT what happens if Tau C is of
51:15older
51:17DT itself
51:22well in this case you see that because
51:25the variance of ETA is of order one over
51:28Tau C it means that beta squared
51:33is 1 over DT
51:37and so ETA is 1 over square root of DT
51:45so if you take the limit where
51:48Tau C is equal to DT or of all the DT
51:51and DT goes to zero
51:53you find yourself with a very thick
51:56differential equation because
51:58ETA is everywhere infinite okay so I
52:02can't even draw this picture it's it's
52:04going to be a mess it's going to you
52:06know be very large positive very large
52:09negative with a zero correlation time so
52:12what does that mean
52:14so in this case you have to you know
52:17work hard mathematically and build
52:20What's called the theory of stochastic
52:23differential equation
52:25so in particular the big name here is
52:27Ito
52:32and the field is stochastic
52:38differential equations
52:49and so you can give a meaning to this
52:51limit
52:52but
52:54in this
52:56framework in this convention
52:59these simple change of variables are no
53:02longer allowed or at least they need to
53:05be supplemented by an extra term
53:08so there's a full story to be told about
53:11uh the ETO term and the correction to to
53:15this change of variable in the equal
53:17convention but I won't speak about this
53:20now I'm just you know pointing this out
53:23to you that in some cases you should be
53:25careful about what convention is more
53:27adapted to your problem
53:29and in many cases when the correlation
53:32time is finite then there's no problem
53:35you should
53:36use the What's called the stratanovic
53:39convention
53:49but in some cases
53:52you should use digital convention so
53:55what are these cases well for example if
53:58your model is actually in discrete time
54:02so if time is not a continuous variable
54:04to start with but it's a discrete time
54:07variable
54:09and that the noise ETA
54:13is always posterior
54:16to
54:18uh
54:20the moment you are now so you never know
54:23what's going to happen in the future
54:25so it's a discrete time step and every
54:28time step J
54:30there's a new thing that's going to
54:31happen in the future and you know
54:33nothing about it
54:34so it means that from one
54:37time step to the next there's no
54:39correlation whatsoever and that you
54:42always before the noise okay
54:45and in this case it turns out that when
54:48you go to the continue limit of this
54:50time step here going to zero
54:54uh you have to use the total convention
54:56okay so I'll I'll show you later in the
54:59lecture a case where it's natural that
55:02you must use The Ether convention
55:05okay so this was a kind of uh
55:08parenthesis
55:10but it's an important one because here
55:12you know already I've done something
55:14that needs to be justified the simple
55:17change of variable if you're not careful
55:19then you're going to do something wrong
55:26so there's a huge number of papers in
55:29the literature about Ito versus
55:31stratanovic and you know when should you
55:33use it and
55:36how to avoid making mistakes I've I've
55:39tried to condense this literature in a
55:42few words and you can have more details
55:45for example in the election notes
55:48uh yes in the lecture notes that the
55:51equality technique wants and the
55:53Valentina is going to speak much more
55:56about this in the city tree
55:59okay so let me go back to
56:02to this result here after having
56:03Justified where it comes from
56:06okay so for Simplicity I'm going to
56:09imagine that uh Z at T equals zero
56:14is equal to one
56:17so u0
56:19is zero
56:21and what I get is that U of T is Mt
56:27Plus
56:28the sum of random terms
56:33they're correlated because they have a
56:35correlation time Tau C but if T is very
56:38large compared to Tau C there's a
56:40central limit theorem that holds even if
56:42the variables are correlated as I told
56:44you the central limit theorem is very
56:46robust so
56:48you don't have to specify uh many things
56:51about the atas you know that provided
56:55they have a second moment which we've
56:56assumed from the start then this thing
56:59for key much greater than Tau C
57:03is going to converge to Mt plus PSI
57:07square root of Sigma t
57:14of Sigma squared t
57:17where xci is
57:20a normal gaussian variable
57:24foreign
57:32okay
57:35so
57:37that's you I know everything about you I
57:40know that if I shift Q by empty then and
57:44divide it by square root of T then what
57:47I get is a gaussian random variable
57:51and so if I knew if I know the
57:52distribution of U
57:55teacher View
57:58then I can reconstruct the distribution
58:00of that ptfz
58:07why because it's just a change of
58:09variable so if I
58:11for all values of U I know the
58:13probability to get a sum U within d u
58:17then I know that it's going to be the
58:19same number of events that contribute to
58:22ptfz within VZ
58:25okay so I can actually write
58:29an equation between the two
58:33which is that all events contributing to
58:36you within the EU are going to be events
58:39that contribute to V within DZ with z
58:44equal exponential View
58:48by the way I'm I'm sloppy in the
58:50notations in the sense that
58:52the these probability distributions I'm
58:55calling them with the same letter
58:58but
58:59I like the convention because it's it's
59:01very useful and I like the convention
59:04like the the oral convention that should
59:07be Rewritten read as the probability of
59:10you and the property of that but of
59:13course T itself is not the same function
59:15okay so it's an abusive notation but I
59:19find it very useful
59:21and usually it's not ambiguous
59:25okay so from this
59:28general rule
59:30which is true when uh it's a monotonic
59:35relation between Zed and you if the
59:38relation is not monotonic it's uh it's a
59:40little more complicated but let me drop
59:43that for the moment because in this case
59:45it's a monotonic relation then from this
59:48relation you immediately get
59:50that Z is distributed according to
59:53What's called the log normal
59:54distribution so PT of Z
59:57is
59:591 over Z
1:00:01which comes from the Jacobian the dudv
1:00:06uh square root of 2 pi Sigma squared t
1:00:12exponential of minus log Z
1:00:16minus m t
1:00:18squared divided by 2 Sigma squared
1:00:22okay and this is called log
1:00:26normal
1:00:28distribution
1:00:31so it's obvious If U is normal
1:00:36um then Zed is the log normal
1:00:40there's a log Z in the in the arguments
1:00:43of the gaussian
1:00:46and as I said there's an extra one over
1:00:47Z which comes from the Jacobian
1:00:51okay so let me say a few things about
1:00:54the log normal
1:00:56because
1:00:57um
1:00:58it is a distribution that is in a sense
1:01:01neither
1:01:03fat tailed nor thin-tailed it's a it's a
1:01:07kind of
1:01:08a statistical monster
1:01:11so of course this is a
1:01:14a description that doesn't mean much but
1:01:17it was the call that way by
1:01:20Bruno Mars
1:01:22and the reason he called it the
1:01:24statistical monster is the following
1:01:27so first of all
1:01:29all the moments of the lognormal are
1:01:32finite
1:01:34so if I compute the average value of Z
1:01:37to the n
1:01:39uh-huh for all for all n
1:01:44larger or equal to zero then zero
1:01:48then you get a finite result
1:01:51and this result is
1:01:54exponential of m t n
1:01:58of nmt
1:02:00Plus
1:02:01N squared Sigma squared T over 2.
1:02:06okay
1:02:08so this is easy to compute from the
1:02:12distribution itself and a few well one
1:02:15gaussian integral
1:02:17so I won't derive it but what's
1:02:20important to note is that you get a
1:02:23finite result for all n
1:02:25and if you remember that's the Criterion
1:02:27I used to call the distribution thin
1:02:30tail
1:02:37but it hide
1:02:39under the rug something that is uh
1:02:42extremely uh nasty about this
1:02:45distribution is that although all the
1:02:48moments are finite
1:02:50you can have a very bad
1:02:52impression about the process if you only
1:02:54focus for example on its average value
1:02:58so usually one thing of fin tail
1:03:00distribution such that if you know about
1:03:04the average value you you know a lot
1:03:06about the distribution
1:03:08think of the exponential for example the
1:03:10exponential that I gave you uh last time
1:03:13well you know the average value and you
1:03:16know that the order of magnitude of all
1:03:18the variables of the exponential
1:03:20distribution are being are going to be
1:03:22the same as that of the average value
1:03:27but imagine that
1:03:29m is negative
1:03:34so I have
1:03:38a log Z here
1:03:40which is U that becomes more and more
1:03:43negative as P increases
1:03:45and therefore Z itself
1:03:50typically should be exponentially small
1:03:52right
1:03:53okay
1:03:55so typically
1:03:57U is equal to minus
1:04:00average absolute value of M times t
1:04:11oops
1:04:12okay I'm sorry I've written too far
1:04:17and nobody shouts it so um
1:04:24someone should stop me
1:04:32right
1:04:33foreign
1:04:41typical
1:04:43is exponential of minus M times t
1:04:48okay so it is true that you know for
1:04:51most
1:04:52realization of PSI
1:04:55U becomes linearly small with t linearly
1:04:59negative with t that's this equation
1:05:02and so Z typically becomes exponentially
1:05:05small in t
1:05:07but if you compute the average value of
1:05:11lead
1:05:14not too well organized here you think to
1:05:17erase this
1:05:31and yeah
1:05:36um
1:05:37so now let's compute average value of Z
1:05:42just by putting n equal 1 in in the
1:05:44general equation and what you find is
1:05:46exponential of minus absolute value of M
1:05:49times t plus Sigma squared T over 2.
1:05:54and so you what you find is if it's
1:05:56Sigma squared over 2 is greater than
1:05:58absolute value of M
1:06:00then average value of Z increases
1:06:03exponentially
1:06:06uh
1:06:10with time
1:06:13so you're in a situation where
1:06:16most probably you're going to observe
1:06:19exponentially small values
1:06:21but if you compute the average you're
1:06:23going to find an exponentially large
1:06:25average
1:06:27so it's a very bizarre situation and how
1:06:31why is it the case well if I plot
1:06:34P of Z as a function of v for T large
1:06:40in this case in this special case what
1:06:43you're going to see is something that's
1:06:46Heidi Peak
1:06:51around
1:06:53the typical value of d
1:06:56so this is exponential of minus absolute
1:06:59of M
1:07:00p
1:07:02so this moves
1:07:04you know the closer and closer to zero
1:07:05it kind of collapses to zero at the next
1:07:07financial speed
1:07:09but there's such a fat tail here
1:07:15that
1:07:16the average value of Z
1:07:18is actually going the other direction so
1:07:22this is moving in that direction whereas
1:07:25the red curve the red line is moving in
1:07:28that direction
1:07:30so it's it's a really weird situation
1:07:36so although
1:07:40the log normal has all its moments and
1:07:43although it decays faster than any power
1:07:46law
1:07:47it looks like a parallel
1:07:51in the sense that you can rewrite the
1:07:53log normal distribution
1:07:56so
1:07:58write it here log log normal
1:08:03in details
1:08:09then you can rewrite mathematically P of
1:08:12Z
1:08:13tfd
1:08:16as
1:08:18in details that going to Infinity
1:08:211 over Z to the 1 plus mu of Z
1:08:26with mu of Z
1:08:30equal so what I'm writing here is is is
1:08:33exact and then I'm going to come in log
1:08:36Z over 2 Sigma squared t
1:08:41minus M over Sigma squared
1:08:45so this is just the rewriting of the log
1:08:48normal in the tail but to make you
1:08:51realize that it looks like a parallel
1:08:54why does it look like a parallel it
1:08:56looks like a parallel because actually
1:08:58log Z is a very slow function of Z so
1:09:02for larger intervals and these intervals
1:09:06become larger when T increases you have
1:09:09the impression that
1:09:12the the there's a power law there's a
1:09:15well-defined parallel regime where in
1:09:18log log this would look straight but
1:09:20actually if you go very very far you
1:09:23recover the fact that it drops faster
1:09:25than a parallel and you recover the fact
1:09:28that all the moments are in the finite
1:09:30so you can be fooled by lognormals
1:09:33because like normals
1:09:35look like power laws although they are
1:09:37not and you know if you want to use a
1:09:41log normal you should be sure that the
1:09:43description is valid in the whole regime
1:09:46otherwise you may you may badly estimate
1:09:49this moment anyway so I wanted to point
1:09:52this out that the log normal is already
1:09:54something that's
1:09:55not a broad distribution but uh
1:09:59a statistical monster in that sense
1:10:03okay
1:10:07so now
1:10:08foreign
1:10:17let me go back to my
1:10:20problem
1:10:25I've dropped the index I
1:10:34because what I said applies
1:10:37for any eye the same
1:10:39math
1:10:42and what I want to explain to you now is
1:10:45that within this model there is
1:10:48a concentration transition
1:10:59so as you see it's a very simple model
1:11:01in the sense that all the cities are
1:11:04growing independently of one another
1:11:06or all the wealth of the individuals are
1:11:08growing independently of one another
1:11:10and let me write again in small here the
1:11:13equation that I'm considering am I that
1:11:17I
1:11:17Plus
1:11:19a to I
1:11:21said I
1:11:23so that's my
1:11:25equation for each eye
1:11:27so this is an equation that you know
1:11:29evolves independently for for each
1:11:31individual or each City but still
1:11:34there's something interesting that's
1:11:36going to
1:11:37take place
1:11:39and the object I'm going to consider is
1:11:44nearly Z
1:11:47which is defined as the sum from I equal
1:11:511 to n
1:11:53of these
1:11:55capital z
1:11:56I
1:11:58so what is the interpretation of curly Z
1:12:01well it's simply the total population of
1:12:04the country
1:12:05assuming that everybody lived in Us in
1:12:08in a city
1:12:09or is the total wealth of the population
1:12:11or
1:12:13if you have other examples in mind then
1:12:15it's uh whatever other examples
1:12:20yes you feel my history on my board
1:12:26Okay so
1:12:29what is going to happen to this object
1:12:32here
1:12:35it's a sum of random variables
1:12:38because the z i is are random through
1:12:41the randomness of the atas
1:12:44and furthermore the z's are independent
1:12:47because okay I haven't
1:12:49that this is an extra assumption I'm
1:12:52assuming that the ati's are independent
1:12:56from I to J so there are no correlations
1:12:59between different cities
1:13:01which may not be true but uh that's the
1:13:04model I'm considering so you have
1:13:06completely independent random variables
1:13:08and I'm summing them so what can happen
1:13:12I'm really in the context of the central
1:13:14limit theorem here but am I in the
1:13:17central limit theorem the classical case
1:13:19or the lady case
1:13:23um
1:13:24well let's let's see so in order to make
1:13:29the discussion slightly simpler to start
1:13:31with I'm going to assume that all the
1:13:34Mis are the same
1:13:37which makes the story even more
1:13:40interesting because in this case
1:13:41everybody on average growth grows at the
1:13:45same speed
1:13:46and so I can rewrite this
1:13:48as
1:13:50uh exponential of Mt
1:13:54thumb from I equal 1 to n
1:13:58of exponential of
1:14:00stigma square root of T times PSI
1:14:06okay
1:14:08the PSI I here is exactly the same PSI
1:14:12as I had here except that now it has an
1:14:15index
1:14:16because for all for different cities or
1:14:20different individuals the realization of
1:14:22the noise is not going to be the same so
1:14:25the atas are not the same and so the PSI
1:14:27I
1:14:28are not the same okay so this is the
1:14:31object I have to deal with
1:14:33and now
1:14:35you you're going to understand
1:14:36immediately that something interesting
1:14:38uh happens
1:14:40by considering two different ways of
1:14:43taking the limits where n goes to
1:14:45infinity and T goes to Infinity so I'm
1:14:48going to consider very large countries
1:14:51and asymptotic times
1:14:53but
1:14:55I can do this in two different ways I
1:14:57can first fix t
1:15:01okay
1:15:02large
1:15:06and take n to Infinity
1:15:10in this case well nothing can happen
1:15:13right because
1:15:15I've told you that this is a log normal
1:15:17distribution
1:15:18so it has all its this this random
1:15:21variable is the distributed according to
1:15:24run log normal distribution rather
1:15:26and so all the moments of this random
1:15:28variable are finite
1:15:31and therefore I'm in the case of the
1:15:33central limit theorem
1:15:35so I have CLT
1:15:38and in particular
1:15:40the haciendo index
1:15:42that is the weight of a of a given
1:15:45country in the whole
1:15:47for the whole for the whole country
1:15:50I I expect that H is of older one over n
1:15:56and this is absolutely true if you take
1:15:58the limit in that way you fix the large
1:16:01but you take n to Infinity that's what
1:16:03you're going to find you're going to
1:16:04find that Curly Z is gaussian
1:16:08and the half interval index is one of
1:16:11rent
1:16:12but there's another limit you could
1:16:14think of and actually there will be a
1:16:18family of ways of taking the limit that
1:16:20I'm going to go to to in a second you
1:16:23could take the limit in the other way
1:16:26around you put 6n
1:16:31large
1:16:34and take T to Infinity
1:16:39but if you do that
1:16:41well
1:16:42clearly because T is going to Infinity
1:16:45you're going to pick up from this sum
1:16:48the largest member
1:16:51because when T goes to Infinity
1:16:54exponential of Sigma square root of T
1:16:58PSI Max
1:17:00which is the largest of all the size
1:17:03appearing here
1:17:05is much bigger
1:17:07than exponential of Sigma square root of
1:17:09t
1:17:10by uh well Max minus one
1:17:17which means the second largest okay
1:17:20because the difference between cymax and
1:17:22cymax minus one is depends on n
1:17:26actually if you remember I told you that
1:17:28if size gaussian this difference
1:17:31actually shrinks with n but it's it's
1:17:34anyway fixed or n fixed and if I take
1:17:37key extremely large at one point
1:17:41the square root of T times PSI Max will
1:17:43be larger than much larger than square
1:17:46root of T times size Max minus 1 and
1:17:48therefore the exponential is going to be
1:17:50even the the difference of scale will be
1:17:54even enhanced and so I can drop all the
1:17:57other terms and approximate curly Z by
1:18:01its largest term
1:18:02okay so in this limit clearly it's
1:18:05dominated by the extreme and the high
1:18:07final index
1:18:09is equal to one
1:18:11or tenth to one
1:18:13okay because as I just said it's going
1:18:16to be the largest
1:18:17term
1:18:19which happens to have been favored by
1:18:21past realization of the noise you see
1:18:24all in in principle all cities grows at
1:18:27the same speed they have all the same Mi
1:18:30but there's one of them that's been
1:18:31luckier in a way
1:18:33and it's going to take it all at the end
1:18:37and so the half handle index tends to
1:18:39one
1:18:40so what happens in the middle
1:18:46well I guess that
1:18:49you've already
1:18:51anticipated what I'm going to say
1:18:53because this this is really an
1:18:55illustration of the Central limit
1:18:58theorem the generalized Central limit
1:19:01theorem is that I can take the limit
1:19:03when n and T Go to Infinity
1:19:06in different fashions
1:19:08and I've shown you two extreme cases
1:19:11but what I'm going to do now is Take n
1:19:15and T Go to Infinity
1:19:18with a fixed mu
1:19:21which of course I'm calling you for a
1:19:22reason which is defined as square root
1:19:25of 2 log n
1:19:28divided by Sigma squared t
1:19:32okay
1:19:34so I'm taking the limit keeping a
1:19:37certain ratio so to say between n and T
1:19:40or rather between log n and T
1:19:42so if I take both to Infinity but fixing
1:19:46log n over t then what you find is that
1:19:50there are three cases
1:19:53either mu is greater than two
1:19:57and you have the usual Central limit
1:19:58theorem
1:20:01so mu greater than 2 means
1:20:03for example fixing T and growing n
1:20:07so it was the first case here fixing T
1:20:09growing n and at one point your mu will
1:20:13be greater than two and uh you'll be in
1:20:16the CLT case
1:20:18but there are the two other cases as
1:20:20well for Mu between one and two
1:20:24you have
1:20:26the levy
1:20:28Central limit theorem
1:20:30and the half single index
1:20:33is
1:20:36um
1:20:37well what I gave you uh a few slides ago
1:20:41so it's n to the 2 1 minus mu over mu
1:20:45sorry
1:20:47um a bad writing but um
1:20:49I've written it before it's exactly the
1:20:51same result
1:20:52and from you
1:20:54lesson one
1:20:56then you have also the levy Central
1:20:58limit theorem for curly Z
1:21:01m u Curry d
1:21:08I'm used as curly Z
1:21:11and the half single index is of all the
1:21:14one
1:21:16and clearly mu lesson one corresponds to
1:21:18the second extreme case
1:21:21here where I Fix N and increase time and
1:21:25then as you see if I do that mu is going
1:21:28to get smaller and smaller and at one
1:21:30point
1:21:31uh we become less than one
1:21:35so this is exactly the same
1:21:36phenomenology as the one I gave you uh a
1:21:39few black balls ago uh the ahafindle
1:21:43index is distributed according to this
1:21:45strange weekly curve
1:21:48and so here what you see is that you
1:21:52have
1:21:54a kind of phase transition in the sense
1:21:56that
1:21:59if I Fix N very large
1:22:04and increased time and I plot the half
1:22:07Intel index
1:22:09then
1:22:10for a long time
1:22:13for the time it takes to reach mu equal
1:22:16one you will have enough internal index
1:22:19that's very small
1:22:21okay that you don't even see and then
1:22:24from a certain critical point onwards
1:22:28the our final index is going to grow
1:22:33linearly and then reach one
1:22:37okay
1:22:43so if you have independently growing
1:22:45processes
1:22:48at the beginning
1:22:50you have a fair amount of equality of
1:22:54democracy
1:22:55but as the time goes on and as
1:22:59Randomness becomes more and more
1:23:01prevalent in the sense that this time
1:23:03becomes more and more relevant at one
1:23:06point you will break this
1:23:09Democratic aspect and you'll find that a
1:23:12few members of the sum
1:23:14are going to contribute uh
1:23:17in an outside fashion okay so it's
1:23:20really what the health index tells you
1:23:22is that you have
1:23:24the localization of
1:23:28the population in some cities or the
1:23:30wealth in some individuals and so on and
1:23:33so forth but what is nice is that this
1:23:37happens in a model where everything is
1:23:40independent
1:23:41okay it's really the natural growth
1:23:44independent growth of each of these
1:23:46variables that at one point makes the
1:23:48whole thing collapse in a few uh in the
1:23:52hands of a few individuals
1:24:02so let me um
1:24:11let me give you a few more details about
1:24:14all this
1:24:16so I haven't given PC here but it's
1:24:18clear what PC is TC is the value of t
1:24:21such that U equal one so TC
1:24:27is equal to
1:24:29um two log n
1:24:34over Sigma squared
1:24:36foreign
1:24:41so by the way you see that
1:24:43because of the login here
1:24:46even if n is very large if N is a
1:24:49million log n is not very large and
1:24:52therefore TC is not very large so very
1:24:54quickly you end up
1:24:56concentrated
1:25:01okay
1:25:06so
1:25:09foreign
1:25:11marks
1:25:27Note One
1:25:29the scenario that I've just outlined
1:25:32in detail for example the value of mu
1:25:35relies on the fact that PSI is gaussian
1:25:47but I put a question mark here because
1:25:49actually you can show that
1:25:52you can have a much broader distribution
1:25:54of of size and still keep exactly the
1:25:58same phenomenology so assume that the
1:26:02distribution of PSI which I call roof
1:26:04PSI
1:26:05decays for PSI going to Infinity as
1:26:09exponential of minus PSI
1:26:11to the s
1:26:14so if s equals 2 is the gaussian case of
1:26:18course
1:26:20but
1:26:21provided s is pretty positive so
1:26:24provided it has this kind of exponential
1:26:27tail with some power
1:26:31then
1:26:32for any s positive you have the same
1:26:35phenomenology
1:26:43so the value of mu is a little different
1:26:46but the the fact that you have this
1:26:49succession of phases between central and
1:26:52ethereum Levy theorem uh delocalized and
1:26:55Levy theorem localized
1:26:58it holds much in a much broader sense
1:27:01than just this uh example that I gave
1:27:04you wax size gaussian
1:27:13now the second remark
1:27:16which is that here I've assumed that all
1:27:19the Mis are equal
1:27:21so in principle
1:27:23uh every everybody is treated on the
1:27:26same ground there's no distinction in
1:27:30terms of uh interest intrinsic quality
1:27:32of the of the city or or you know if you
1:27:37think of wealth there's no you you don't
1:27:40believe that some investors are more uh
1:27:43are smarter than others they have the
1:27:45same expected rate of growth
1:27:49um in these two examples because the Mis
1:27:51are all equal to m
1:27:53but of course you could imagine that M
1:27:55itself
1:27:59Is Random
1:28:02not in time now it's fixed in time but
1:28:05random over individuals
1:28:13so some have a larger m forever and
1:28:16others have a smaller m forever and so
1:28:19if I assume that for example Mis
1:28:24our gaussian
1:28:31with mean M Bar
1:28:34and variance capital Sigma Square
1:28:38okay
1:28:39I'm assuming that Mi is now a random
1:28:42variable fixed in time but random over
1:28:44individual and that the repartition of
1:28:47these M's over different individuals is
1:28:49gaussian then you find again the same
1:28:52phenomenology but what changes is the
1:28:55value of mu
1:28:56U is now given by
1:28:59uh square root of 2 log n
1:29:04divided by capital Sigma t
1:29:11and so in particular the uh
1:29:16condensation transition
1:29:19the localization transition because
1:29:23faster
1:29:24you remember TC was 2 log n over Sigma
1:29:27squared
1:29:29now TC
1:29:33is square root of log n
1:29:37so
1:29:38it's intuitive it's due to the fact that
1:29:42in the first case it was random
1:29:45fluctuations
1:29:46for you know a very equal cities or
1:29:50individuals that led to localization now
1:29:53it's the fact that
1:29:54you know some individuals
1:29:57like by assumptions are growing faster
1:30:00than others and this will lead to this
1:30:03concentration happening before but again
1:30:06you find the same again the same
1:30:09phenomenology
1:30:13good and to finish
1:30:16so for one time on time
1:30:20to finish I'm going to give you an
1:30:22example of exactly what I've said up to
1:30:26now in the context of growth
1:30:28an example coming from physics
1:30:47in this example is called
1:30:50the random energy model
1:31:06it was invented by Bernard
1:31:11in nineteen
1:31:13eighty
1:31:16and it was invented to understand that
1:31:18the problem of so-called spin glasses
1:31:21which I uh maybe speak about a little
1:31:24more later but you can think of that as
1:31:27a model to understand the thermodynamics
1:31:30of uh of of of of random objects of
1:31:34amorphous objects
1:31:37and so
1:31:39remember that
1:31:42the position function of a
1:31:44thermodynamical problem Z
1:31:47is equal to the sum
1:31:49over all configurations
1:31:52of exponential of minus beta
1:31:56e of C
1:32:00so this is the so-called partition
1:32:01function
1:32:07and for those of you who have gone
1:32:10through through statistical mechanic
1:32:12courses you know that
1:32:14beta is the inverse temperature
1:32:17and the partition function contains all
1:32:19the thermodynamics of the system for
1:32:22example the free energy is is related to
1:32:25the log of this partition function
1:32:28but you can also Define the weight
1:32:31of configuration C
1:32:34which is
1:32:35exponential of minus
1:32:40the boltzmann the boltzmann factor
1:32:42divided by capitalism V
1:32:46and this gives you the probability to be
1:32:48in configuration C
1:32:52and there's no reason why you shouldn't
1:32:55think of this problem the same way as we
1:32:58thought about what I talked about before
1:33:00so you can also introduce the halfindl
1:33:04index and the half in the index will
1:33:06just be the sum overall configurations
1:33:08of W Squared of C okay
1:33:14and so again the question will be
1:33:16whether
1:33:18the cell symbol index goes to zero
1:33:22and if it goes to zero it means that the
1:33:24system is exploring over time a lot of
1:33:27different configurations
1:33:29so in a sense it's a liquid
1:33:37okay
1:33:39but if this half indoor index
1:33:42even for large systems tends to a
1:33:44constant
1:33:45greater than zero
1:33:47it means that a substantial fraction of
1:33:50the time the system will be found in
1:33:54only a few configurations
1:33:56okay there's only a handful of
1:33:58configuration that is that are going to
1:34:01contribute
1:34:02significantly to the partition function
1:34:05and in this case
1:34:08you want to call it the glass
1:34:10it's a glass because the system gets
1:34:12stuck in some configuration and not
1:34:16others
1:34:18so at this stage I haven't said anything
1:34:20about
1:34:21the energies of the configuration
1:34:26and what dirida proposed is to think of
1:34:30a random system in a highly simplified
1:34:33manner such that the energy of different
1:34:37configurations are all independent
1:34:40random variables
1:34:41okay
1:34:43so of course this is a enormous
1:34:45approximation
1:34:46because you know if you think of a real
1:34:49glass if you change a little bit the
1:34:51positions of the molecule the energy
1:34:53won't change much so there's no real
1:34:56deep
1:34:57um a trivial at these reasons to believe
1:35:00that at some site that there might be
1:35:03other ways to think about the problem
1:35:05which actually show that derida got it
1:35:08right but that's much beyond what I want
1:35:10to tell you today
1:35:11the only thing I want to tell you is
1:35:14that as soon as you make this virida
1:35:16assumption that the energies are
1:35:19independent random variables then what
1:35:21you see is that the problem I'm
1:35:23considering here
1:35:25is formally analogous to the problem I'm
1:35:29I consider above
1:35:33and so okay one has to get the scaling
1:35:35right I've swept under the rugs the
1:35:38dependence on the size of the system of
1:35:41the energies and the number of
1:35:43configurations
1:35:44which play the role of uh t and n if you
1:35:47want but if you
1:35:49formulate the problem in a natural way
1:35:52what you find is that the Berita random
1:35:56energy problem
1:35:58has exactly the same phenomenology as
1:36:00the one I gave above and in particular
1:36:04now what plays the role of time
1:36:06is inverse temperature and so what you
1:36:10find is that there's a critical
1:36:12temperature TC
1:36:14which is 1 over beta C
1:36:17such that if T is greater than t c
1:36:22the handle index goes to zero with the
1:36:25size of the system
1:36:26and so the system is indeed a liquid
1:36:29and if T is less than t c
1:36:33then the half angle index
1:36:36the average up in the index is given by
1:36:381 minus U you remember so it's 1 minus t
1:36:41over t
1:36:44and so you get a glass
1:36:46but if you go into the mathematical
1:36:49details of the model you realize that
1:36:52it's exactly what I told you up to now
1:36:54so for free in a sense we've solved the
1:36:56random energy model
1:36:58so of course here I have not
1:37:00distinguished between mu greater than 2
1:37:03and U less than two
1:37:06and what happens is that there's an
1:37:08intermediate temperature regime between
1:37:09TC and 2tc where
1:37:13um something uh you know intermediate
1:37:16happens the the speed at which
1:37:18uh the half signal index goes to zero is
1:37:21is anomalous but apart from that there's
1:37:24nothing much different between the two
1:37:27temp the temperature regime so usually
1:37:29one does not distinguish the two and
1:37:31just focus on the on the glass
1:37:34transition so PC is the glass transition
1:37:39foreign
1:37:45the freezing transition is really a
1:37:48transition of condensation in Phase
1:37:51space
1:37:53so that's the idea freezing
1:37:58is equal to condensation
1:38:02of the boltzmann weight
1:38:04in Phase space
1:38:13so although what I told you today seemed
1:38:16very far from physics to start with in
1:38:19the end we recover a very interesting
1:38:21example of this condensation transition
1:38:24in the context of solid state physics
1:38:28so that's what I wanted to tell you
1:38:29today
1:38:31um I hope I was uh clear and not too
1:38:34fast again it's very very difficult to
1:38:37know whether you're you know you're
1:38:39following what I'm saying so maybe I can
1:38:42take a few questions now
1:38:48hello
1:38:53is anyone still there
1:38:57yes
1:39:00okay
1:39:03um
1:39:03any question any remark
1:39:06complaints
1:39:08but yes I have a question
1:39:10yes yes I don't fully understand uh on
1:39:13the other Broadway the Criterion for the
1:39:15different phases uh the new
1:39:19I don't fully understand why it's
1:39:21expression comes from with the square
1:39:23root of 2 again and
1:39:25Sigma Square C
1:39:32you're asking where this equation comes
1:39:34from yes
1:39:36yes okay yes well
1:39:39as I said I mean what I'm trying to do
1:39:42is to find a balance between going into
1:39:45much details over the mathematics and
1:39:48giving you the story so if you want to
1:39:51derive this thing
1:39:54what you should do is to use uh the
1:39:59extreme value statistics
1:40:01that is something that I've alluded to
1:40:03last time the distribution of the
1:40:07largest variables okay and if once one
1:40:11know how the one knows how the how the
1:40:14size in the extreme
1:40:16behave
1:40:18then it's it's not very complicated to
1:40:21show that the distribution of this
1:40:23object here
1:40:27is a parallel
1:40:29and the exponent of the parallel is
1:40:31given by this so one has to do a little
1:40:34bit of computation but essentially what
1:40:37happens and you'll find in the notes is
1:40:39that for a broad family of distributions
1:40:42and actually this corresponds to the
1:40:44family I've written here
1:40:46uh the tail
1:40:48of the distribution of the maximum
1:40:51events
1:40:52of these ones is exponential
1:40:56and this exponential tail is independent
1:40:58of on the value of s okay
1:41:02and so what that's what's interesting is
1:41:04that once you have an exponential
1:41:06distribution for the size
1:41:08in details
1:41:10you have a parallel distribution for
1:41:12exponential API and the exponent is
1:41:15given by that so maybe it's worth me
1:41:18giving you something that will help you
1:41:22thinking about these problems
1:41:25and I'm not sure where we speak about
1:41:26this particular case but I think oh okay
1:41:28so you so you have that in a okay you
1:41:31have that you know today but this is
1:41:33what I call the Battle of exponentials
1:41:35so if you have a a random variable X
1:41:39which is such that it decades is
1:41:41exponential of minus Lambda X
1:41:45but you're looking at Y which is
1:41:47exponential of beta X
1:41:49okay so you have
1:41:52variables that have a very very small
1:41:54probability to be large but you're
1:41:55looking at the exponential of these
1:41:57variables then it's easy to show and
1:41:59you'll see that with Valentina that P of
1:42:01Y is 1 over y to the one plus mu
1:42:05with mu equals uh uh Lambda over beta
1:42:13okay so you'll see that in detail but in
1:42:16details but what I want to say is that
1:42:18this argument is really behind the way
1:42:21you go from
1:42:23this thing to Mu okay
1:42:26okay so you can try to do it if you want
1:42:28but uh I otherwise we I can help you but
1:42:32what I think I can help you
1:42:33okay thank you
1:42:39okay
1:42:41so for those of you who want to leave
1:42:44and make a pause before Valentina starts
1:42:47at 11
1:42:48I can stay 10 more minutes if you want
1:42:51and
1:42:52interact with you
1:42:58so I guess you see my screen okay
1:43:03my
1:43:04Blackboard okay and what I see
1:43:10yes sorry
1:43:12hello yes
1:43:14yes the immune question is keptical
1:43:17instant while you
1:43:19you keep increasing nmt up to Infinity
1:43:23I'm sorry I it's very hard to hear you
1:43:27your mind doesn't seem to you're all
1:43:29muffled
1:43:31yes maybe I'm sorry is it okay
1:43:36maybe you can type your question then
1:43:59so the MU is kept constant yes exactly
1:44:03exactly this is the point this is
1:44:06this is how
1:44:09you know coming back to here I I gave
1:44:12you two extreme cases where you take n
1:44:15and T to Infinity but T6 very large and
1:44:18going to infinity and the other case
1:44:20here
1:44:21and I said there's an intermediate
1:44:23regime between the two and of course the
1:44:26intermediate regime can be parametrized
1:44:27by a lot of different ways but the
1:44:31correct way to parametrize it to get
1:44:33something non-trivial is to introduce
1:44:36this object here
1:44:37and now take both n and T to Infinity
1:44:41at the sixth rate if you want between
1:44:45the two and this is the proper if you
1:44:49want a mathematical way to actually get
1:44:52to these results these results hold
1:44:56mathematically in the limit I said so
1:44:59you need both t and n to be large
1:45:02but
1:45:03keeping this fixed but of course in
1:45:06practice
1:45:07and that's what I try to allude to Here
1:45:09by making this little graph in practice
1:45:11you'll never be in that mathematical
1:45:14limit but you'll be in a limit where n
1:45:16is large and T grows and if you're in in
1:45:20that case where you can't really take
1:45:22that limit with this fixed then you have
1:45:25the phase transition as a function of T
1:45:27but
1:45:29then if everything is finite and not
1:45:32infinite all these statements become
1:45:34approximate I mean they've come and
1:45:37they're not asymptotic rigorous
1:45:39statements they're just
1:45:40approximate descriptions of what's going
1:45:43on okay
1:45:59um I have a question for the random
1:46:00energy model so you mentioned that the
1:46:03first sight seems to be a naive model
1:46:06and you said that if you look a bit more
1:46:08closely it's a it's a good model can you
1:46:12maybe give just a little explanation why
1:46:14it would be a good model
1:46:18you mean the random energy model
1:46:20yes
1:46:22yes okay so
1:46:25the idea
1:46:27that makes it
1:46:29a good model in the end is is far from
1:46:33trivial actually
1:46:35and it's
1:46:40it came in a sense as a surprise in the
1:46:43sense that you can
1:46:45write models which are which have a
1:46:49microscopic formulation in terms of
1:46:52spins if you want
1:46:53so these are real degrees of freedom
1:46:56with the real hamiltonian coupling these
1:46:58fins and then you make a complicated
1:47:01calculation
1:47:02and what you end up with is something
1:47:05that has exactly the phenomenology that
1:47:07I've just uh
1:47:09given to you
1:47:12so in a sense the microscopic model you
1:47:15started from
1:47:17appears to map
1:47:19exactly onto the random energy model
1:47:22in some limit in a way of thinking about
1:47:25it and so the reason is that although
1:47:28individual configurations cannot be
1:47:31considered as random and so this is a
1:47:34very bad approximation
1:47:36as independent what happens is that if
1:47:39you group configurations together and
1:47:41make bundles of configurations then
1:47:44these bundles of configurations can be
1:47:47considered as independent so let me let
1:47:50me make a little drawing of
1:47:53the landscape of the energy of of these
1:47:56random systems
1:47:58so of course it's a it's a very bad
1:48:01drawing because configuration space is
1:48:04the kind of infinite dimensional object
1:48:07and and I'm drawing a one-dimensional
1:48:10function so you know it's it's a very
1:48:13very bad
1:48:15picture of what's going on in infinite
1:48:18dimensional space spaces
1:48:20but what you know in words what happens
1:48:24is that the energy of this configuration
1:48:26and of this configuration they can't be
1:48:29considered to be independent they're
1:48:31very close by and so it's stupid to
1:48:34think that they're independent energy
1:48:36but what happens physically is that the
1:48:39system is going to be stuck in in this
1:48:42Valley here so this this blob of
1:48:45configuration holds the system for a
1:48:48certain time and so there's kind of
1:48:50local equilibration within the valleys
1:48:54and now if you cause grain the system at
1:48:57the scale of the valleys and think not
1:49:00of the individual Energies
1:49:03the configuration of individual uh
1:49:06the individual configuration energies
1:49:09but rather the free energy of these
1:49:12blobs then it's a much better
1:49:15approximation to think that these are
1:49:17independent random variables and then
1:49:20what I told you the note one that I
1:49:23erased is that it doesn't depend much
1:49:26what the tail of the distribution of
1:49:28these energies is really because in the
1:49:32end it's all going to map to the same
1:49:34model
1:49:35and this is due to the universality of
1:49:37the statistics of extremes if you want
1:49:39so any
1:49:41distribution that falls as exponential
1:49:43of minus size yes if you want is going
1:49:46to be in the same universality class so
1:49:49there's a there's a conspiration of many
1:49:52subtle points here that in the end makes
1:49:55the make the random energy model not
1:49:58such a bad model after all and I haven't
1:50:01given you uh some some of the properties
1:50:04of the random energy model from the
1:50:06thermodynamical properties but you know
1:50:09as a kind of bare bone model for glasses
1:50:12it's surprisingly good in view of
1:50:16uh the information that you've put in
1:50:18the model which is close to zero okay
1:50:21so that's that's the story in a nutshell
1:50:26thank you
1:50:28okay I think it's time to for me to
1:50:30leave the floor to
1:50:33Valentina
1:50:34I'm going to erase my board and see
1:50:38yeah yeah sure and you need it
1:50:59foreign
1:51:34foreign
1:52:12foreign
1:52:46this conference will now be recorded
1:52:49very good and can you see the Blackboard
1:52:50because maybe using this
1:52:52color which is lighter is not great
1:52:54right
1:52:57it's okay
1:52:59okay great
1:53:01okay so welcome everybody to uh the
1:53:03first today as I told you several times
1:53:06my name is Valentin and this is the
1:53:08email where you can find me for any
1:53:11questions or comments or whatever you
1:53:13want to tell me
1:53:15so uh there are several things that I
1:53:17wanted to mention before we start this
1:53:19today so the first one is about the
1:53:22mailing list
1:53:23so you should have received an email
1:53:25either on Wednesday evening or during
1:53:28the weekend stating that you are
1:53:30included in the main English and with
1:53:33some information about the course in the
1:53:34today so in case you have not received
1:53:37this email please either write your
1:53:40email address on the chat or send me an
1:53:42email so that I can add you to the list
1:53:44and this is important because uh we will
1:53:47send information about the course in the
1:53:49today using that mailing list without
1:53:51going through
1:53:52the main list of the full and the same
1:53:55Master course
1:53:57and the second thing that I promised to
1:54:00discuss with you today is about the
1:54:01duration of the city
1:54:03and the fact is that we have one hour in
1:54:06principle so that they should be from uh
1:54:0811 to 12.
1:54:11but there are some today which are a
1:54:13little bit uh longer and we saw last
1:54:16year that this time was not enough to
1:54:18discuss let's say all of the exercises
1:54:22so for this reason this year well first
1:54:24of all I'm giving to you the text of the
1:54:27today in advance and the idea is not
1:54:29that you solve the exercises before we
1:54:31discuss them but it is more that you
1:54:34look at the tax so there will be in
1:54:36future so there's some problems or
1:54:38exercises which are formulated in a
1:54:40language that is a little bit different
1:54:42so one has to familiarize
1:54:45uh a bit so there's somebody who doesn't
1:54:47hear
1:54:49uh
1:54:50is it a problem
1:54:54Yes means that you hear me right
1:54:59yes okay
1:55:04okay
1:55:06so sorry about the problem and in case
1:55:08feel free to speak so so that I can see
1:55:11these and more directly than on the chat
1:55:15so as I was saying the idea is that
1:55:18maybe it would be good if you look just
1:55:20at the text of the today a bit in
1:55:21advance so that we spend less time
1:55:24during this hour to discuss the
1:55:27framework
1:55:28and on top of that what we can do is to
1:55:31make uh we study the session a little
1:55:33bit longer so let's say until uh 12 15
1:55:36or 12 30 but this depends uh on you so
1:55:40there might be people that have courses
1:55:42after this there might be people that
1:55:44have constraints there might be people
1:55:46that find it very tiring to stay online
1:55:48all of the time so uh so what I thought
1:55:52about doing is
1:55:54um is a little bit it's a little survey
1:55:56that we can do maybe at the end of this
1:55:58today so it will give you a link to our
1:56:00website and you just have to click
1:56:02whether you would prefer to have it one
1:56:04hour long or one hour and a half and
1:56:06then based on the average we can decide
1:56:09that unless there is somebody who tells
1:56:11me that there are sharp constraints uh
1:56:14and that we do have to finish at 12
1:56:16because you have maybe other courses
1:56:18afterwards so if that is the case let me
1:56:21know now otherwise we do this little
1:56:23survey at the end
1:56:25okay so what is the plan of the uh the
1:56:28today of today so this one is gonna be
1:56:31actually a little bit shorter I think
1:56:33because it might be about things that uh
1:56:37everybody here has seen uh at least once
1:56:40in her or his life uh in the sense that
1:56:44what we are going to do today is discuss
1:56:46uh some basic stuff about probability in
1:56:49particular uh this formula of change of
1:56:52variables that have been uh that popped
1:56:55out in the lecture also today and we are
1:56:58doing this with a perspective that is
1:57:00the one of uh Power laws since this has
1:57:03been discussed a lot in the last lecture
1:57:05and in this lecture so the idea is to
1:57:07try to see how these power laws emerge
1:57:10and what are possible mechanism to get
1:57:12this type of distributions and just to
1:57:15give an idea of uh of the first three
1:57:18tables this will be more generally about
1:57:20probability probability like today a
1:57:22little bit of statistics uh in the next
1:57:25day so the next one will be about uh
1:57:27maximum likelihood and there will be
1:57:29also statistics and the kolmogorov
1:57:32smirnoffstash in particular in in the
1:57:34last two days the number nine and then
1:57:36the third today is about stuff that has
1:57:39been introduced today so Ito versus
1:57:41strathanovic and langevani in general
1:57:44stochastic calculus so this is a pretty
1:57:46long uh today but we will go back to
1:57:49these issues when discussing a
1:57:51particular example in uh into the seven
1:57:54so that's a bit uh the plan for the
1:57:57first uh three weeks and now let me
1:58:00start with uh with the problem of today
1:58:03so I think and I hope you all have the
1:58:05text of the today which was uh in the
1:58:08ens folder
1:58:11but just in case so let me recap very
1:58:16fast what is the theory that is uh
1:58:19behind all of the exercises so the idea
1:58:21is just following we have some random
1:58:23variables that I call capital X
1:58:31that takes values let's say in R so you
1:58:35have a density function for this random
1:58:38variable that I call rho of X and in
1:58:41here in this today I'm using this
1:58:43subscript to indicate what is the random
1:58:45variable of which I am Computing the
1:58:48density function and of course what's in
1:58:49parenthesis is the value that I assign
1:58:52to the function so this is uh density
1:58:55and let me also introduce the so-called
1:58:58cumulative
1:59:00that's the function which is what tells
1:59:03you what is the probability that your
1:59:04random variable takes values which are
1:59:06let's say smaller or equal to this small
1:59:09X in here so you get it integrating
1:59:12from let's say minus infinity or
1:59:15whatever is the lower edge of the
1:59:17support of your distribution up to the
1:59:19point x your density function evaluated
1:59:23at y so this is a
1:59:26cumulative density function
1:59:29another problem of today is the
1:59:30following we have a new random variable
1:59:33y that is related to X through some uh
1:59:37functional relations or through some
1:59:39function G and we want to understand
1:59:42what is the distribution of Y given that
1:59:44we know the distribution of x
1:59:47and there is a simple formula that was
1:59:50written also today in the lecture before
1:59:52at the Blackboard which assumed that g
1:59:55is monotonic
1:59:59and this formula tells you that you get
2:00:02the distribution of Y simply taking
2:00:05the density
2:00:07of your original variable X and dividing
2:00:10it by the absolute value
2:00:12of the derivative of the function that
2:00:15gives you the relation between these two
2:00:17random variables here and what you have
2:00:20to do is to compute this at X which is
2:00:23the inverse through your functional
2:00:25relation of the point Y of which you
2:00:28want to compute the density
2:00:31so this is the very basic formula that
2:00:34is more or less everything we will need
2:00:36for for the exercises of today
2:00:40uh okay so let me give you a little
2:00:44trick
2:00:45to remember this so of course one way to
2:00:48remember this is to remember what is the
2:00:51formula for the change of variables of
2:00:52the data function that you can use to
2:00:55relate the two random variables but here
2:00:57let me give you another trick which is
2:01:00more let's say probabilistic in flavor
2:01:03and the idea is to derive this using
2:01:06this cumulative density function so
2:01:08suppose that we have a function G
2:01:12but this is my ex
2:01:14this is G of X and we have a function
2:01:16which is monotonic and I take it to be
2:01:19increasing for example
2:01:21and I want to compute so I fix one value
2:01:24of y
2:01:26which corresponds to one and only one
2:01:28value of x because the function is
2:01:30monotonic
2:01:31and what I want to do is to compute
2:01:35what is the probability that my random
2:01:39variable X is smaller or equal to some
2:01:43given value of x and because you have
2:01:45this one-to-one relationship you see
2:01:47that you can write this so this is
2:01:49nothing but the f of x which I just
2:01:51introduced
2:01:52and what you can do is to write it so as
2:01:56asking that X is smaller than this
2:01:58particular value corresponds because of
2:02:00this one-to-one relationship to having Y
2:02:02which is smaller than this particular
2:02:05value which is nothing but the function
2:02:07G evaluated at X so what this means is
2:02:10that this probability here is exactly
2:02:13equal to the probability that Y is
2:02:15smaller or equal to this particular
2:02:17value which is G of x
2:02:21and once you have this basic
2:02:23relationship of course in here I'm
2:02:25assuming the G is an increasing function
2:02:31G was decreasing what he would have had
2:02:34instead of this speed would be 1 minus
2:02:37this probability right because the
2:02:39relationship is inverted but once you
2:02:42have this it is very easy to get out
2:02:44this formula the only thing that you
2:02:46have to do is to take the derivatives of
2:02:48this Expressions because this
2:02:51probability in here is now the
2:02:53cumulative density function of your
2:02:55variable y evaluated at G of x
2:02:59and now if you take the derivative of
2:03:01the cumulative what you get out of it is
2:03:04just the density of your random
2:03:06variables and so taking the derivative
2:03:08of this expression you would get
2:03:10that rho X of X is what is
2:03:15the derivative of this evaluated at G of
2:03:17X so this is
2:03:19the density of Y evaluated as G of x
2:03:23times the derivative of the argument
2:03:25which is exactly
2:03:27G Prime of x
2:03:29and so you see that you have to bring
2:03:31this on the other side you replace G of
2:03:34X with Y and then X will be given by G
2:03:36to the minus 1 of Y and this is how you
2:03:39get this formula in here
2:03:43okay so uh this is simple let me just
2:03:46tell you about two
2:03:49generalizations before we go to the
2:03:52exercises
2:03:58foreign
2:04:02and of course feel free to ask questions
2:04:04or make comments you can unmute you
2:04:09I think that's faster
2:04:13okay so generalization
2:04:21so the first one is
2:04:23of course whenever you have a function
2:04:25so as you see here I assumed that the
2:04:28function is monotonic so either
2:04:29increasing or decreasing but you may
2:04:32have functional relationship G which are
2:04:35non-monotonic
2:04:37foreign
2:04:45over all of the possible values of X
2:04:49which correspond to the particular value
2:04:51of y that you have chosen so your
2:04:53density
2:04:55will be now given by a sum overall point
2:04:58x i such that
2:05:01G at x i is equal to
2:05:04your particular value of y of exactly
2:05:07the same thing that you see up there row
2:05:10X evaluated at X PSI divided by G Prime
2:05:14evaluating the text time and if you want
2:05:17to check this expression you may use so
2:05:21I will not do the example in here maybe
2:05:23to share a little bit of time but you
2:05:25may use for instance
2:05:29simple example like the function x
2:05:31square and you will see that using the
2:05:33trick of the cumulative density
2:05:35functions you get exactly this formula
2:05:37in here
2:05:39stop me if you want me to to do this
2:05:43exercise of course we can do it
2:05:44and the second generalization is if you
2:05:47are in higher dimensions
2:05:54so you may want to transform variables
2:05:58which are not one number but which are
2:06:00vectors for instance so let's say that X
2:06:03has
2:06:05n components
2:06:07and from X you want to Define some other
2:06:11Vector y
2:06:13such that is each component is a
2:06:15function
2:06:17of all of the components of the vector X
2:06:19so we'll have G1 G2
2:06:22up to
2:06:24GN of x
2:06:28and now if you go higher a dimensional
2:06:30what you have to do to adapt this
2:06:32formula in here is to replace
2:06:35the
2:06:37um
2:06:37absolute value
2:06:39of the derivatives that you add in there
2:06:42with the absolute value of the
2:06:45determinant
2:06:47of a matrix and this Matrix is the
2:06:50so-called Jacobian of the change of
2:06:52variables so the Matrix
2:06:55J has components i j
2:06:58that are what so this will be the
2:07:01derivative
2:07:02of the if component of your vector y
2:07:07over the JS component of of your
2:07:11original random variable X
2:07:14and if you want to practice with this
2:07:15there is the bonus exercise
2:07:20of the today which is exactly
2:07:23which is exactly an example of how to
2:07:27use this formula
2:07:29okay so given this so you will see that
2:07:33there is then a simple example which is
2:07:35solved in the text which is about this
2:07:37log normal distribution that had just
2:07:40been discussed uh in the lecture so I
2:07:42will not go through that again
2:07:45and I will jump
2:07:47to the first exercise
2:07:51that is about
2:07:54how to practice with this equation here
2:08:01and how to understand simple mechanisms
2:08:04by which power load distributions appear
2:08:11[Music]
2:08:16any question
2:08:18no
2:08:21okay so let's go
2:08:24through exercise one
2:08:28so exercise one starts like this you are
2:08:31given a random variable
2:08:33X which has a uniform distribution let's
2:08:36say between
2:08:38minus zero and one
2:08:42so the density
2:08:44of my random variable X I can write it
2:08:47as
2:08:48Theta of X so this is a function which
2:08:51is uh which takes value one only when
2:08:54the argument is larger or equal to zero
2:08:56and then I have Theta 1 minus X which
2:09:00tells me that my variable has to be
2:09:02smaller than one
2:09:04and the first thing that you're giving
2:09:06is a variable Y which is minus Lambda
2:09:11log of x with
2:09:14Lambda positive and we have to compute
2:09:17what is the distribution of Y so here
2:09:19you see you have a functional relation
2:09:21between X and Y you have a g of X which
2:09:24is monotonic this is just minus Lambda
2:09:28log of x
2:09:30so we just have to apply the formula
2:09:32that I gave before and do a little bit
2:09:34of algebra so from here we have the G
2:09:36Prime of x
2:09:38is
2:09:40minus Lambda over X if we invert this
2:09:44relation so if we put this equal to Y
2:09:46and we invert it we get that X is equal
2:09:49to e to the minus y over Lambda
2:09:53and given this
2:09:55is what we can say is that that
2:09:58is then that rho y of Y is
2:10:03what it's rho of X so Theta of X beta of
2:10:071 minus X
2:10:09divided by the absolute value of this uh
2:10:12first derivative so Lambda is positive
2:10:14so I just have a factor of Lambda in
2:10:16here and then in the numeral in the
2:10:18numerator yes I would have absolute
2:10:20value of x but before I have because I
2:10:22have this Theta I just I can just put an
2:10:25X in here and I have to compute it
2:10:28at e to the minus y over Lambda
2:10:32so what I get out of this is a factor of
2:10:36e to the minus y over Lambda divided by
2:10:38Lambda
2:10:39and then you see so these Theta is
2:10:43asking that my exponential is positive
2:10:46and this is true for every value of y
2:10:48whereas this Theta is asking that this
2:10:51exponential is smaller than one and in
2:10:52order to add this I have to impose that
2:10:56Y is positive
2:10:58so what I obtain is is an exponential or
2:11:02LaPlace
2:11:03distribution
2:11:09I see a question
2:11:14okay the left
2:11:18okay maybe I'll try and write a little
2:11:21bit more in the center
2:11:24okay so the logarithm of a uniform
2:11:26distribution is uh is an exponential
2:11:29distribution and now let me go to the
2:11:31second point
2:11:34uh okay
2:11:44foreign
2:11:57or the Blackboard this time
2:12:00so the second point is okay now you're
2:12:03given another random variable Z which is
2:12:06e to the beta times this variable y of
2:12:10which we had just computed the
2:12:12distribution and I didn't write it on
2:12:14the text but I assumed here that beta is
2:12:17is larger than zero and so now how is
2:12:20this type distributed well you know it
2:12:23from the last five minutes of the of the
2:12:25lecture what you find is that this is uh
2:12:29okay let me do this maybe it's faster so
2:12:32what is g of
2:12:33why now is e to the beta y
2:12:37so G Prime of Y
2:12:40beta into the Beta y
2:12:43and what is y is
2:12:46log of Z over beta so using this
2:12:52the density of our random variable Z
2:12:55will be what so you you use the same
2:12:58formula as before and what you should
2:13:00find is that we can write this and write
2:13:04this density as new
2:13:06fita
2:13:09of Z minus 1 divided by Z
2:13:13as the one plus Nu
2:13:16where mu
2:13:17in this example is given by 1 over
2:13:22beta Lambda so you have a factor of
2:13:24Lambda which comes from the exponential
2:13:26distribution of your variable Y and then
2:13:28you have a factor of beta which comes
2:13:30from this functional relation so in the
2:13:33lecture this formula was given except
2:13:36that Lambda was replaced by 1 over
2:13:39Lambda but it is exactly the same thing
2:13:41so this is a Pareto
2:13:43or power law distribution
2:13:51and the mechanism is what we said before
2:13:54so we have a random variable whose
2:13:57distribution is suppressed exponentially
2:13:59so it is exponentially unlikely to find
2:14:02very large values of this variable y but
2:14:06what you do in here is to enhance
2:14:09uh this these values of Y exponentially
2:14:13through this relation so you see that
2:14:15this is like the partition function of
2:14:17this random manager model and this is
2:14:19the inverse temperature so y would be
2:14:21the energy that here we are taking
2:14:23exponentially distributed and if you do
2:14:26this combination of exponentials so the
2:14:28exponential of a variable which is
2:14:30exponentially distributed what you get
2:14:32out is precisely a power row
2:14:34distribution
2:14:36with an exponent that depends on uh on
2:14:39the properties of both your original uh
2:14:43exponential distribution and how does it
2:14:46depend on Lambda well the idea is that
2:14:48you see when mu is smaller the Tails of
2:14:53your distribution are enhanced so you
2:14:55have a larger probability
2:14:57to get large values of your random
2:15:01variables that and these happen when mu
2:15:03is molar which means that either beta or
2:15:05Lambda are are large and this makes
2:15:09sense because if you increase Lambda
2:15:10what you are doing through your
2:15:12exponential distribution is giving more
2:15:15weight
2:15:16or more probability two variables or
2:15:19values of Y which are larger so it makes
2:15:21sense that you get a larger value of
2:15:23that and at the same time if you also
2:15:26increase beta what you're doing is
2:15:27giving more weight to this large random
2:15:30variables so again it makes sense that
2:15:33these two couplings enter in your
2:15:35exponent through this relationship
2:15:39okay so there is another comment that we
2:15:41can do or make starting from this
2:15:45which is essentially related
2:15:482.3 and the comment is that so what is
2:15:51this set is e to the beta y but y we
2:15:55derived its distribution assuming that Y
2:15:58is given by minus Lambda a log of x and
2:16:01x is uniformly distributed so we can
2:16:04also write
2:16:05Z as
2:16:07e to the minus beta Lambda
2:16:10log of x
2:16:12so this is 1 over X
2:16:14to the power of beta Lambda
2:16:17so another way to interpret the
2:16:19emergence of this power flow is to say
2:16:22that we are given a distribution that is
2:16:26uniform so which has some flat density
2:16:29for instance around zero and then we
2:16:32take the reciprocal or a negative power
2:16:35of this distribution and what we get out
2:16:37is a power law of distribution that
2:16:39gives a big weight to values which are
2:16:42last
2:16:43and this is because we have a finite
2:16:45weight in our original distribution for
2:16:47values which are small because our
2:16:49distribution was uh was flat and uniform
2:16:53around zero
2:16:54and so this observation is essentially
2:16:56the idea of uh
2:16:58we can make it more general and this is
2:17:01the idea of the point three
2:17:03so the point three
2:17:07says suppose that you have a random
2:17:11variable X
2:17:12whose distribution
2:17:14products uh evaluated attacks is let's
2:17:19say positive when evaluated at zero
2:17:24and we also assume that that this
2:17:28function rho of X is somehow smooth
2:17:30around zero so we can have a Taylor
2:17:31expansion so this means that
2:17:35my distribution will look something like
2:17:37this around the x equal to zero
2:17:42and then we can ask what is the
2:17:43distribution of any negative power of my
2:17:46variable X so let me introduce y now as
2:17:49X
2:17:51to the minus gamma
2:17:53and let's use again so let's practice
2:17:56again with this formula of change of
2:17:58variables so this is uh this is my G so
2:18:02now G Prime of x
2:18:05is minus gamma x to the minus 1 minus
2:18:09gamma
2:18:10and what is X as a function of Y well X
2:18:14is y to the minus
2:18:171 over gamma okay
2:18:20so what will be our density I hope you
2:18:23can see it
2:18:25our density
2:18:27of our new variable y so this will be
2:18:31our row X
2:18:33evaluated at X where X is now 1 over y
2:18:39to the one over gamma and then you have
2:18:42this G Prime so gamma
2:18:44let's assume that gamma is positive
2:18:47so the absolute value gives me gamma and
2:18:51then I would have uh what I get in here
2:18:53is a factor of Y to the one plus gamma
2:18:59divided by gamma
2:19:02okay
2:19:03and so what you see is that when y
2:19:05becomes large so if you look somehow at
2:19:08the Tails of uh of your distribution and
2:19:12one over y goes uh close to zero so we
2:19:15can approximate this density to the
2:19:17value that it has in zero we can do a
2:19:20Taylor expansion if you want and keep
2:19:21only the first coefficient in the
2:19:24expansion which we assume to be
2:19:26different from zero so this will be row
2:19:28of X evaluated at zero and then we have
2:19:32below gamma y to the one plus
2:19:36one over gamma so this uh at large
2:19:40values of Y will look like operator
2:19:43distribution with mu equals to 1 over
2:19:45gamma plus Corrections which come from
2:19:48the higher order expansion of your
2:19:50distribution of the variable X
2:19:53and this is in line with uh with what we
2:19:56found for this particular shape of the
2:19:59distribution
2:20:01okay so to summarize we have two
2:20:04mechanisms to easily produce power load
2:20:07distribution one is this combination of
2:20:09exponentials that we will discuss also a
2:20:12little bit more in the second day and
2:20:15the other one is to take some negative
2:20:17power or the reciprocal of of a
2:20:21distribution which is flat around zero
2:20:25Okay so
2:20:27having practice a little bit with this
2:20:29formula Let Me Now go to uh to the
2:20:32second exercise
2:20:33which is an exercise that is a little
2:20:35bit related to the homework so
2:20:39let me keep this I don't know if you had
2:20:41time to have a look at the arm works
2:20:45and by the way if there are any problems
2:20:47with setting up
2:20:49the
2:20:51python or Jupiter
2:20:55just let me know we can discuss it at
2:20:58the end of the today's session
2:21:04foreign
2:21:08so I will keep this and I will rewrite
2:21:11our change environment no okay well yes
2:21:15let me rewrite it
2:21:21just as a reminder
2:21:24our formula of change of variables
2:21:31okay and then we go to this exercise too
2:21:34so the ideal VM work for those uh who
2:21:37had no time to uh to look at it was to
2:21:40discuss a little bit with real data the
2:21:43emergence of the so-called deep low
2:21:46which is something that is very robustly
2:21:49emerges anytime you look at
2:21:51distributions in the size of cities for
2:21:55examples or distributions in the
2:21:57frequency of worth which is the example
2:21:59that we discussed in the homework and
2:22:02the idea is the following so you have a
2:22:06text in the homework I took the example
2:22:09of the book by James Joyce
2:22:12Ulysses or Ulysses and what you can do
2:22:16is to sample to take all of the words
2:22:18which appear in the text and to order
2:22:20them as a function of their frequency of
2:22:23appearance in the text and so in this
2:22:26way what you do is you associate which
2:22:29word a rank that goes from one to the
2:22:32total number of words that you find so
2:22:35the most frequent word the one which
2:22:38appears more often in the text will have
2:22:40rank equal to one and what you plot in
2:22:43here is the frequency
2:22:44of the word as a function of the rank so
2:22:47this is the number of times
2:22:52the word appears
2:22:57and you can normalize it by the total
2:23:00number of words in your text and what
2:23:03you find is that of course this is a
2:23:04decreasing function because this is the
2:23:06definition of the rank so the word of
2:23:10rank 1 which Ulysses is I think uh the
2:23:14word the
2:23:15is the one which appears with higher
2:23:17frequency and then you will have the
2:23:19second most frequent word the third one
2:23:21and so on
2:23:22and what you find several times in this
2:23:25type of real data is that the
2:23:27distribution
2:23:29follows somehow a power law so you see
2:23:32that
2:23:33F of R goes like 1 over R to some power
2:23:37Alpha and the exponent is very close to
2:23:40one
2:23:41and this is what it's called Uh trip
2:23:44flow so this tells you that the most
2:23:46frequent word appears twice as often as
2:23:49the second most frequent word and so on
2:23:51and so forth
2:23:52so the idea of uh the second part of the
2:23:55armor kind of these exercises to try to
2:23:57give a little simple probabilistic model
2:24:00for this type of power row and this is
2:24:04done introducing a model of random
2:24:06language so uh
2:24:15in this random language what you do is
2:24:17this is the problem of monkey with a
2:24:20typewriter this is how it is it called
2:24:22so there is that you have somebody who
2:24:24types randomly on a keyboard and it
2:24:27creates a text and the keyboard has a
2:24:30space and it has a certain number of
2:24:32letters which I call M in the exercise
2:24:35and you have a certain probability to
2:24:37hit the space so let's say the space is
2:24:41heated with a probability that I call Q
2:24:46okay let me write it as Rob
2:24:49of heating the space is equal to q and
2:24:52then the letters
2:24:54you can hit them with equal probability
2:24:57randomly so the probability of each
2:24:59letter will be
2:25:021 minus 2 divided by the total number of
2:25:05letters that you have which is given by
2:25:08n
2:25:09and what is a word so in this way if you
2:25:12type randomly you will create strings of
2:25:14letters plus spaces and we identify
2:25:17words that I will call
2:25:20Omega with an index I and I goes from 1
2:25:23to Infinity so the words are strings of
2:25:25letters plus a space at the end which
2:25:28tells me that the word is over
2:25:31okay so what we want to understand is uh
2:25:34what should we expect if we did a plot
2:25:36like this for this example of random
2:25:38language
2:25:40and let's do this following
2:25:43the exercise so there is a first
2:25:46question which is about
2:25:49probability of getting some particular
2:25:52words so this point one
2:25:56based on this model so the idea is that
2:25:59we can ask now we select one particular
2:26:02word
2:26:03Omega I which could be for example I
2:26:06don't know acdf
2:26:08uh hear the word as a length which is
2:26:11equal to four and I can ask what is the
2:26:14probability to get to observe in my
2:26:17random text this particular word on the
2:26:20guide
2:26:20and there is probability you can compute
2:26:22it easily so now let's assume that the
2:26:24length of the word is is small L what
2:26:28you find is that the probability does
2:26:30not depend on the particular word that
2:26:32you're looking at but only on the length
2:26:33so on the number of letters which you
2:26:36have because letters are also equally
2:26:38probable so this is
2:26:39uh given by Q which is the probability
2:26:42to have the space at the end
2:26:45times the probability to have four
2:26:47letters which is 1 minus 2 divided by m
2:26:51to the power of the number of letters
2:26:53that is the length of the word
2:26:56okay
2:26:58so this gives you the probability of
2:27:01having one particular word of length l
2:27:03so let me put a small line in here
2:27:06indicating that you have a dependence on
2:27:08this variable and once you have this you
2:27:11can ask well what is the probability
2:27:12that I
2:27:14encounter any word of length l
2:27:18so no matter what is the particular
2:27:20choice of the letter I can ask how many
2:27:22times should I find Words which have
2:27:25length equal to four in my uh in my
2:27:28random uh text and this is again easily
2:27:32derived from this because what you have
2:27:34to do is to multiply the probability of
2:27:36a word of length L times the number of
2:27:39words that you can create of length L
2:27:41and how many are this word well for each
2:27:45position in here you can choose any
2:27:48letter out of the N uh ones that you
2:27:51have in your keyboard and so this
2:27:53probability of length will be M to VL
2:27:56which is the number of words of length L
2:27:58times the probability of one of them
2:28:04and so you see that this if I interpret
2:28:06length as a random variable I have
2:28:08something which is exponentially
2:28:09distributed so this is Q E to b l
2:28:13log
2:28:16of 1 minus Q where my Lambda in the
2:28:20notation of before is now equal to minus
2:28:231 over log of 1 minus Q
2:28:28okay so I hope this is uh clear
2:28:33and you can check that this is well
2:28:35normalized so if you sum over all
2:28:36possible values while you get that this
2:28:38probability is equal to one
2:28:41okay now we have the probability of
2:28:43length and the next thing uh that we
2:28:46want to try to do is to uh somehow
2:28:49understand what is the behavior of the
2:28:51rank and to do this we want to derive a
2:28:53relationship between the rank that we
2:28:56should expect for uh for our random
2:28:59words and their length of which we know
2:29:02the distribution in here
2:29:04and what I mentioned this is something
2:29:06that if you
2:29:08did the exercise in the homework you may
2:29:11have observed is that in this type of
2:29:14random language if you try to redo
2:29:17a plot like this you find
2:29:20something that is apparently a little
2:29:22bit funny so you find some degeneracies
2:29:25in the frequency of words so you find
2:29:27that you have a set of words that has
2:29:29more or less
2:29:31uh the same frequency which are the most
2:29:34frequent then you have another bunch of
2:29:36words
2:29:37which I've more or less the same
2:29:40frequency smaller than the first one and
2:29:42so on and so forth and of course what we
2:29:44want to do is to derive let's say the
2:29:47continuous or any an interpolation for
2:29:50this data arranged in this way so why do
2:29:54we have this designer thing well this
2:29:55comes from the fact that all of this
2:29:57words of the same length are equip
2:30:00probable so you have essentially the
2:30:02same probability to find the word the
2:30:05and the word ABC in this type of random
2:30:08text and this is what gives you
2:30:10frequencies that are very similar and
2:30:13what you then expect is that all of
2:30:15these groups are distinguished by the
2:30:18length of the words and that belongs to
2:30:21the group and so the idea is that words
2:30:25which have length let's say equal to one
2:30:27so a single letter they are much more
2:30:30probable than longer words because the
2:30:32distribution is exponential so they will
2:30:34occupy the first values of the rank
2:30:36because they will appear more frequently
2:30:39so they will have values of the rank
2:30:41which belong to the interval
2:30:451 up to the total number of words of
2:30:49length L equal to one that is simply
2:30:51given by n
2:30:53so stop me here if if this reasoning is
2:30:56not clear
2:30:57but if you understood it then it's very
2:31:00easy to uh then identify that the second
2:31:03set of words will have uh length two and
2:31:05the rank will go from n plus one to M
2:31:10plus
2:31:12M squared which is the number of uh the
2:31:15total number of words of length two and
2:31:17so on and so forth so using this
2:31:20reasoning we can now ask
2:31:23what is the rank of a word of or the
2:31:27interval to which the rank of a word of
2:31:30length l arbitrarial
2:31:33belongs to
2:31:40let me check time okay
2:31:46so let me call it
2:31:48the rank of a particular word
2:31:51let me say Omega I of length L well this
2:31:55will belong to some interval which has
2:31:58lower bound that is given by the sum of
2:32:01all of the ranks of the words with
2:32:04shorter lines so this will be sum from K
2:32:07from 1 to
2:32:10L minus 1
2:32:12of the number of words which I have to
2:32:15allocate in to get the rank which is m
2:32:20to the k for each value of the length
2:32:22and in here I will have uh to place
2:32:27so I have the same thing but I have to
2:32:29place also the words of length equal to
2:32:31the one that I'm looking at
2:32:33so this will be something like this
2:32:37so is this reasoning uh clear if this is
2:32:40not clear please ask questions
2:32:43or we can discuss this more but if this
2:32:46is clear then we can easily do the sums
2:32:49in here so we get that
2:32:52therefore this rank
2:32:57so these sounds we can do explicitly we
2:33:00get something like m one minus m
2:33:041 minus m to the power L and in here we
2:33:07have the same except that we have
2:33:09n minus 1 instead of L
2:33:16and out of this relationship or bound
2:33:19what we can reduce or what we can set a
2:33:23little bit as an approximation in the
2:33:25sense that now we assume that somehow
2:33:27the variable R is a continuous variable
2:33:30we want to functionally relate it to uh
2:33:33to the length L and this bounce tell us
2:33:36that we can assume that the rank as a
2:33:39function of the length
2:33:40goes like or let me put an equal here
2:33:45some constant which depends on N that we
2:33:48are not really interested in but then we
2:33:50are interested in the dependence on L
2:33:52and this is of the type e to the L
2:33:56log of n
2:33:58so essentially the rank goes like M to
2:34:01VL
2:34:02up to Corrections which depend on my
2:34:05constant m
2:34:07okay so this gives us a relationship
2:34:09between the variable of which we know we
2:34:12want to compute the distribution because
2:34:14this frequency plot is essentially
2:34:16giving us the distribution of the rank
2:34:19interpreted as a random variable and
2:34:21another random variable which is the
2:34:23length of which we know the distribution
2:34:25because we know that this is exponential
2:34:28and so what we have is precisely this
2:34:30mechanism of a combination of
2:34:33exponentials so let me use the notation
2:34:36that we introduced in the first exercise
2:34:41so there we add Z which was equal to the
2:34:45beta Y and we had a p of Y
2:34:48which was e to the minus y over Lambda
2:34:52and this was giving us an exponent that
2:34:55is Mu equal to 1 over beta Lambda
2:34:58for the reasons that we have explained
2:35:00so what does this correspond to in our
2:35:03example so y now is played by L and so
2:35:08instead of Lambda what we have is minus
2:35:13one over log of 1 minus Q
2:35:19and instead of beta
2:35:22there there it is so this is now our new
2:35:25functional relationship and our beta is
2:35:27is what is log of n
2:35:34and therefore the exponent mu that we
2:35:37should expect for the distribution of
2:35:38the rank
2:35:40is is what is minus
2:35:45log of
2:35:471 minus Q
2:35:49divided by
2:35:51log of n
2:35:54right
2:35:56and so what we said is that
2:35:59in this
2:36:00import
2:36:02probabilistic
2:36:03example of random language F of R would
2:36:07go like 1 over r
2:36:11to the one minus
2:36:13logo
2:36:161 minus Q divided by sorry I hope you
2:36:20see it
2:36:21log of n
2:36:24and in front you have some constant
2:36:26which depends on M and which depends on
2:36:28Q
2:36:31oh you see that
2:36:37before you see that the correction to
2:36:39one which would be your deep flow is is
2:36:42particularly small and therefore if your
2:36:45alphabet is sufficiently large you see
2:36:48that you have the emergence of of a
2:36:50power law with the exponent that is very
2:36:53similar to the exponent that you see in
2:36:55in the examples with with real text
2:36:59okay so this uh concludes a little bit
2:37:02uh the idea of the exercise and there
2:37:05was just one last comment and maybe we
2:37:07can if we have five minutes we can look
2:37:10at the homework uh together so the last
2:37:13comment is that of course
2:37:15this example is very simplified so real
2:37:19languages are not random languages and
2:37:22there is one particular let's say
2:37:24assumption that we are making which I I
2:37:28think is particularly false for the case
2:37:31of real languages so can anybody guess
2:37:34uh what is the thing that perhaps we
2:37:37shouldn't expect if you are dealing with
2:37:39real text
2:37:42and letters
2:37:45oh I don't hear you the microphone is
2:37:48really bad
2:37:50right now
2:37:56no I don't hear you anymore
2:38:04okay maybe you can type uh
2:38:08it's true so the fact that letters are
2:38:10equivalent is uh is more or less exactly
2:38:13related to what I wanted to say and
2:38:15because it what gives you
2:38:19this Factor here so what we are saying
2:38:22is that any combination of letters is a
2:38:25word and so the number of letters of
2:38:27words sorry that we can have increases
2:38:30exponentially with the line and of
2:38:32course in real language this is not true
2:38:34so it's not true that any arbitrary
2:38:37combination of letters is a word and
2:38:38therefore it is not obvious that you
2:38:40should expect that the number of
2:38:42distinct words that you have of a given
2:38:45length is uh satisfies the scaling and
2:38:48this is something that uh you you will
2:38:51see in the exercise playing with this
2:38:54now I don't know if I if we can
2:38:57maybe spend
2:39:00try to do some
2:39:10Helen is
2:39:16let me just catch one minute
2:39:20and commentary on the homework
2:39:30so on the homework you will see
2:39:32first of all that if you use a random
2:39:35language you do see a deep plot which
2:39:38looks like this so with all of this and
2:39:41the generatives but uh let's say Trend
2:39:44that is of the form one over R and this
2:39:47is 0.7
2:39:48okay
2:39:51of the homework
2:39:56and then I also asked at a certain point
2:39:58to plot the histogram of the number of
2:40:01distinct words that you find in your
2:40:03text as a function of the length and to
2:40:05plot it in log scale so what you find so
2:40:09this is let's say the length and this is
2:40:12the number
2:40:14of this import the log
2:40:18and indeed for random text what you
2:40:20should find is is that you recover as
2:40:24you shoot this exponential Behavior at
2:40:27least for the small enough values
2:40:28available and then you have that your
2:40:30curve bends and the reason why it bends
2:40:32is that words with very large lengths
2:40:35are really improbable because the
2:40:38probability decays exponentially so you
2:40:41have to create a text which are
2:40:43extremely large in order to detect them
2:40:45with their full statistics so the reason
2:40:47why this distribution Banks and you
2:40:49don't see the school and in our behavior
2:40:51is that you have a cutoff which is given
2:40:54by the length of the text that you are
2:40:56generating
2:40:57that does not allow you to somehow
2:40:59sample the tales of this exponential
2:41:03distribution and if you try to do so
2:41:05this is for random text
2:41:08and if you try to redo this for Ulysses
2:41:11so for real tax you see that this is not
2:41:13quite the case right you should get an
2:41:15histogram that goes something like that
2:41:17well not linear much or curved uh it's
2:41:22just
2:41:23a text and this is precisely related to
2:41:26uh to this comments that letters are
2:41:28nothing we probable and not all
2:41:30combinations of letters are uh taking
2:41:33Awards in uh in real language
2:41:37okay so that's it so now I think
2:41:41uh so as you see today yesterday was
2:41:43short and we are right sitting in one
2:41:46hour but uh maybe now we can take two
2:41:49minutes or some minutes
2:41:51to discuss a little bit about uh the
2:41:54lengths
2:41:55and what you would like to do
2:41:58and so anybody has some opinion let's
2:42:01say strong opinion
2:42:04about what's best to do
2:42:10if not I'll try to make
2:42:13a survey with you
2:42:19let me check if this works
2:42:23so I just want to do a survey online so
2:42:25that we are all we all know the results
2:42:29if you go to the website that I'm giving
2:42:32you
2:42:35uh say the time
2:42:41try this it's the first time I try this
2:42:44system so I'm not even sure that this
2:42:46works
2:42:48[Music]
2:42:49um
2:42:55but I think you should find a window
2:42:57where you just have to choose
2:42:59sorry you're here where you just have to
2:43:01choose what you prefer so
2:43:05so we more or less see
2:43:09and something I should specify is that
2:43:11in any case I will upload solutions to
2:43:14all of the exercises so even if we do
2:43:16not have time to discuss everything and
2:43:19the Blackboard you have the solutions
2:43:20there is this question and answer file
2:43:22where we can ask and discuss a little
2:43:25bit more if there are things which are
2:43:27not clear
2:43:28and uh and so even if we choose one hour
2:43:33perhaps
2:43:35there is also a way for you to to look
2:43:39at the rest of the material but one hour
2:43:41and a half is a bit better I think
2:43:44a bit less frustrating
2:43:53and excuse me
2:43:55yes uh can you just show the upper board
2:43:58please
2:44:00the one which is below
2:44:02[Music]
2:44:04yes
2:44:06thank you
2:44:07okay great
2:44:15okay so it seems that there is
2:44:19some preference
2:44:21for making it one hour and a half it's
2:44:2579 of you
2:44:28so I think that we can do that so
2:44:31starting from next time
2:44:32there are four people missing that
2:44:38she left vote okay I will give you the
2:44:40results the full results once uh this is
2:44:44over but perhaps starting from next time
2:44:45as you will see that the day will be a
2:44:47little bit longer uh the next one and
2:44:50the third one in particular so maybe
2:44:51it's good to have one hour and a half
2:44:53and then if we go on and we see that
2:44:56there are shorter Swan of course uh we
2:44:58can make everything uh in one hour
2:45:02okay so is there any question was uh
2:45:05everything okay any questions also on
2:45:07the homework I don't know how much time
2:45:08you have to do uh the arm work but
2:45:12um if you have troubles
2:45:15just let me know now or we can
2:45:19discuss using the email
2:45:31and you could read everything the speed
2:45:33was okay any comments
2:45:36foreign
2:45:47[Music]
2:45:50okay very well thank you guys
2:45:53uh okay so yeah it definitely looks like
2:45:57uh we can go to uh one hour and a half
2:46:01so that's what we will try to do next
2:46:03Wednesday
2:46:04okay
2:46:05thanks to everybody and again uh feel
2:46:08free to write in this question and
2:46:10answer five six it does not need to be
2:46:13scientific questions you can also ask or
2:46:15say comments about the structure of the
2:46:18lectures or problems about exercises
2:46:21homeworks and everything and I will
2:46:22upload right now the solutions to this
2:46:25today and homework and the text for the
2:46:27next day in homework and we see each
2:46:30other next Wednesday
2:46:33so have a nice week everybody
2:46:36thank you bye
2:46:38thank you thank you bye

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